Saturday 04 Apr 2010

University of Palestine

Computer Science II

Trees

Trees

bull Introductionbull The Trees Abstract Date Typebull Basic Algorithms on Trees

Outline

Trees

bull Productivity experts say that breakthroughs come by thinking ldquononlinearlyrdquo

bull Trees as a nonlinear data structure gives ways of accessing and finding elements of better performance than linear structures such as sequences

Introduction

Trees

bull The main terminology for tree data structures comes from family trees with the term ldquoparentrdquo ldquochildrdquo ldquoancestorrdquo and ldquodescendentrdquo

Introduction

Trees

Example Prophets tree [1]

سيدنا على صلي اللهممحمد

[1] httpimg246imageshackusimg24635828618qz1png

The Trees Abstract Data Type

A tree is an abstract data type that stores elements hierarchally With the exception of the top element each element in a tree has a parent element and zero or more children

A tree is usually visualized by placing elements inside ovals and rectangles and by drawing the connections between parents and children with straight lines

Definition

root

The Trees Abstract Data Type

The tree T is a set of nodes storing elements in a parent-child relationship with the following properties

bull T has a distinguished node r called the root of T that has no parent

bull Each node v of T distinct from r has a parent node ubull For every node v of T the root r is ancestor of vbull If a node u is the parent of node v then we say that v is a child of ubull Two nodes that are children of the same parent are siblingsbull A node is external or leave if it has no childrenbull A node is internal if it has one or more children

Terminology and Basic properties

The Trees Abstract Data Type

Terminology and Basic properties

root

Parent of I J K

children of F

The Trees Abstract Data Type

Terminology and Basic properties

Descendant of B A

Descendants of F B and A

The Trees Abstract Data Type

Terminology and Basic properties

Ancestor of all other nodes

Ancestor of I J K

The Trees Abstract Data Type

A tree T is ordered if there is a linear ordering defined for the children of each node in such a way we can identify the children of a node as being the first second third and so on Ordered trees typically indicate the linear relationship existing between siblings by listing them in a sequence or enumeration in the correct order

Terminology and Basic Properties

The Trees Abstract Data Type

Terminology and Basic Properties

Book

Preface Part A Part B References

P1 P2 CH 1 Ch2 Ch3 Ch4 Ch5 P1

The Trees Abstract Data Type

A binary tree is a tree with the following propertiesbull Each internal node has at most two children (exactly two for properbinary trees)bull The children of a node are an ordered pairbull We call the children of an internal node left child and right childbull Alternative recursive definition bull A binary tree is either1048708 a tree consisting of a single node or1048708 a tree whose root has an ordered pair of children each of which is a

binary tree

Terminology and Basic Properties

The Trees Abstract Data Type

Example of a binary tree

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applicationsDecision trees

Binary tree associated with a decision process1048708 internal nodes questions with yesno answer1048708 external nodes decisionsExample dining

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applications

Binary tree associated with an arithmetic expression1048708 internal nodes operators1048708 external nodes operandsExample arithmetic expression tree for theexpression (2 times (a minus 1) + (3 times b))

Terminology and Basic Properties

The Trees Abstract Data Type

Generic methods

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()

Accessor methods

Position root()Position parent(Position p)Enumeration children(Position p)

Query methods

boolean isInternal(Position p)boolean isExternal(Position p)boolean isRoot(Position p)

Update method

Object replace (Position p Object o)void swap(Position v Position w)

Trees ADT methods

The Trees Abstract Data Type

A Tree Interface in Java

PositionalContainerltltinterfacegtgt

SimpleTreeltltinterfacegtgt

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()bull Object replace (Position p Object o)bull void swap(Position v Position w)

bull Position root()bull Position parent(Position p)bull Enumeration children(Position p)bull boolean isInternal(Position p)bull boolean isExternal(Position p)bull boolean isRoot(Position p)

Basic Algorithms on Trees

Depth and Height

bull The depth of a node v is the number of ancestors of v excluding v itself

bull If v is the root then the depth of v is 0bull Otherwise the depth of v is one plus the depth of the parent v

Basic Algorithms on Trees

Depth and Height

public int depth(SimpleTree TPosition v)

if(TisRoot(v))

return(0)

else

return(1+depth(Tparent(v)))

Basic Algorithms on Trees

Depth and Height

bull The height of an entire tree T is defined to be the height of the root of T Another way to view the height of a tree T is the maximum depth of a node of T which is achieved at one or more external nodes

Basic Algorithms on Trees

Depth and Height

public int height1(SimpleTree T)

int h

Enumberation nodes_of_T=Tpositions()

while(nodes_of_ThasMoreElements())

Position v=(Position)nodes_of_TnextElement()

if(TisExternal(v))

h=Mathmax(hdepth(Tv))

return h

The Trees Abstract Data Type

Example

Height Algorithm 1

Basic Algorithms on Trees

Depth and Height

public int height2(SimpleTree T Position v)if(TisExternal(v))return 0elseint h=0Enumeration children_of_v= Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()h=Mathmax(hTheight2(w))return 1+h

Preorder Traversal

Definition

bull A traversal of a tree T is a systematic way of accessing or ldquovisitingrdquo all the nodes of T

bull In a preorder traversal of a tree T the root of T is visited first and then the subtree rooted at its children are traversed recursively

Preorder TraversalAlgorithm

Algorithm preorder(Tv)Visit(v)for each child w of v dopreorder(Tw)

Preorder TraversalPreorder Print

public void preorderPrint(SimpleTree TPosition v)Systemoutprintln(Telement(v))Enumeration children_of_v=Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()preorderPrint(Tw)

Postorder TraversalDefinition

bull Recursively traverses the subtrees rooted at the children of the root first and then visits the root

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29

Trees

bull Introductionbull The Trees Abstract Date Typebull Basic Algorithms on Trees

Outline

Trees

bull Productivity experts say that breakthroughs come by thinking ldquononlinearlyrdquo

bull Trees as a nonlinear data structure gives ways of accessing and finding elements of better performance than linear structures such as sequences

Introduction

Trees

bull The main terminology for tree data structures comes from family trees with the term ldquoparentrdquo ldquochildrdquo ldquoancestorrdquo and ldquodescendentrdquo

Introduction

Trees

Example Prophets tree [1]

سيدنا على صلي اللهممحمد

[1] httpimg246imageshackusimg24635828618qz1png

The Trees Abstract Data Type

A tree is an abstract data type that stores elements hierarchally With the exception of the top element each element in a tree has a parent element and zero or more children

A tree is usually visualized by placing elements inside ovals and rectangles and by drawing the connections between parents and children with straight lines

Definition

root

The Trees Abstract Data Type

The tree T is a set of nodes storing elements in a parent-child relationship with the following properties

bull T has a distinguished node r called the root of T that has no parent

bull Each node v of T distinct from r has a parent node ubull For every node v of T the root r is ancestor of vbull If a node u is the parent of node v then we say that v is a child of ubull Two nodes that are children of the same parent are siblingsbull A node is external or leave if it has no childrenbull A node is internal if it has one or more children

Terminology and Basic properties

The Trees Abstract Data Type

Terminology and Basic properties

root

Parent of I J K

children of F

The Trees Abstract Data Type

Terminology and Basic properties

Descendant of B A

Descendants of F B and A

The Trees Abstract Data Type

Terminology and Basic properties

Ancestor of all other nodes

Ancestor of I J K

The Trees Abstract Data Type

A tree T is ordered if there is a linear ordering defined for the children of each node in such a way we can identify the children of a node as being the first second third and so on Ordered trees typically indicate the linear relationship existing between siblings by listing them in a sequence or enumeration in the correct order

Terminology and Basic Properties

The Trees Abstract Data Type

Terminology and Basic Properties

Book

Preface Part A Part B References

P1 P2 CH 1 Ch2 Ch3 Ch4 Ch5 P1

The Trees Abstract Data Type

A binary tree is a tree with the following propertiesbull Each internal node has at most two children (exactly two for properbinary trees)bull The children of a node are an ordered pairbull We call the children of an internal node left child and right childbull Alternative recursive definition bull A binary tree is either1048708 a tree consisting of a single node or1048708 a tree whose root has an ordered pair of children each of which is a

binary tree

Terminology and Basic Properties

The Trees Abstract Data Type

Example of a binary tree

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applicationsDecision trees

Binary tree associated with a decision process1048708 internal nodes questions with yesno answer1048708 external nodes decisionsExample dining

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applications

Binary tree associated with an arithmetic expression1048708 internal nodes operators1048708 external nodes operandsExample arithmetic expression tree for theexpression (2 times (a minus 1) + (3 times b))

Terminology and Basic Properties

The Trees Abstract Data Type

Generic methods

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()

Accessor methods

Position root()Position parent(Position p)Enumeration children(Position p)

Query methods

boolean isInternal(Position p)boolean isExternal(Position p)boolean isRoot(Position p)

Update method

Object replace (Position p Object o)void swap(Position v Position w)

Trees ADT methods

The Trees Abstract Data Type

A Tree Interface in Java

PositionalContainerltltinterfacegtgt

SimpleTreeltltinterfacegtgt

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()bull Object replace (Position p Object o)bull void swap(Position v Position w)

bull Position root()bull Position parent(Position p)bull Enumeration children(Position p)bull boolean isInternal(Position p)bull boolean isExternal(Position p)bull boolean isRoot(Position p)

Basic Algorithms on Trees

Depth and Height

bull The depth of a node v is the number of ancestors of v excluding v itself

bull If v is the root then the depth of v is 0bull Otherwise the depth of v is one plus the depth of the parent v

Basic Algorithms on Trees

Depth and Height

public int depth(SimpleTree TPosition v)

if(TisRoot(v))

return(0)

else

return(1+depth(Tparent(v)))

Basic Algorithms on Trees

Depth and Height

bull The height of an entire tree T is defined to be the height of the root of T Another way to view the height of a tree T is the maximum depth of a node of T which is achieved at one or more external nodes

Basic Algorithms on Trees

Depth and Height

public int height1(SimpleTree T)

int h

Enumberation nodes_of_T=Tpositions()

while(nodes_of_ThasMoreElements())

Position v=(Position)nodes_of_TnextElement()

if(TisExternal(v))

h=Mathmax(hdepth(Tv))

return h

The Trees Abstract Data Type

Example

Height Algorithm 1

Basic Algorithms on Trees

Depth and Height

public int height2(SimpleTree T Position v)if(TisExternal(v))return 0elseint h=0Enumeration children_of_v= Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()h=Mathmax(hTheight2(w))return 1+h

Preorder Traversal

Definition

bull A traversal of a tree T is a systematic way of accessing or ldquovisitingrdquo all the nodes of T

bull In a preorder traversal of a tree T the root of T is visited first and then the subtree rooted at its children are traversed recursively

Preorder TraversalAlgorithm

Algorithm preorder(Tv)Visit(v)for each child w of v dopreorder(Tw)

Preorder TraversalPreorder Print

public void preorderPrint(SimpleTree TPosition v)Systemoutprintln(Telement(v))Enumeration children_of_v=Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()preorderPrint(Tw)

Postorder TraversalDefinition

bull Recursively traverses the subtrees rooted at the children of the root first and then visits the root

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29

Trees

bull Productivity experts say that breakthroughs come by thinking ldquononlinearlyrdquo

bull Trees as a nonlinear data structure gives ways of accessing and finding elements of better performance than linear structures such as sequences

Introduction

Trees

bull The main terminology for tree data structures comes from family trees with the term ldquoparentrdquo ldquochildrdquo ldquoancestorrdquo and ldquodescendentrdquo

Introduction

Trees

Example Prophets tree [1]

سيدنا على صلي اللهممحمد

[1] httpimg246imageshackusimg24635828618qz1png

The Trees Abstract Data Type

A tree is an abstract data type that stores elements hierarchally With the exception of the top element each element in a tree has a parent element and zero or more children

A tree is usually visualized by placing elements inside ovals and rectangles and by drawing the connections between parents and children with straight lines

Definition

root

The Trees Abstract Data Type

The tree T is a set of nodes storing elements in a parent-child relationship with the following properties

bull T has a distinguished node r called the root of T that has no parent

bull Each node v of T distinct from r has a parent node ubull For every node v of T the root r is ancestor of vbull If a node u is the parent of node v then we say that v is a child of ubull Two nodes that are children of the same parent are siblingsbull A node is external or leave if it has no childrenbull A node is internal if it has one or more children

Terminology and Basic properties

The Trees Abstract Data Type

Terminology and Basic properties

root

Parent of I J K

children of F

The Trees Abstract Data Type

Terminology and Basic properties

Descendant of B A

Descendants of F B and A

The Trees Abstract Data Type

Terminology and Basic properties

Ancestor of all other nodes

Ancestor of I J K

The Trees Abstract Data Type

A tree T is ordered if there is a linear ordering defined for the children of each node in such a way we can identify the children of a node as being the first second third and so on Ordered trees typically indicate the linear relationship existing between siblings by listing them in a sequence or enumeration in the correct order

Terminology and Basic Properties

The Trees Abstract Data Type

Terminology and Basic Properties

Book

Preface Part A Part B References

P1 P2 CH 1 Ch2 Ch3 Ch4 Ch5 P1

The Trees Abstract Data Type

A binary tree is a tree with the following propertiesbull Each internal node has at most two children (exactly two for properbinary trees)bull The children of a node are an ordered pairbull We call the children of an internal node left child and right childbull Alternative recursive definition bull A binary tree is either1048708 a tree consisting of a single node or1048708 a tree whose root has an ordered pair of children each of which is a

binary tree

Terminology and Basic Properties

The Trees Abstract Data Type

Example of a binary tree

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applicationsDecision trees

Binary tree associated with a decision process1048708 internal nodes questions with yesno answer1048708 external nodes decisionsExample dining

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applications

Binary tree associated with an arithmetic expression1048708 internal nodes operators1048708 external nodes operandsExample arithmetic expression tree for theexpression (2 times (a minus 1) + (3 times b))

Terminology and Basic Properties

The Trees Abstract Data Type

Generic methods

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()

Accessor methods

Position root()Position parent(Position p)Enumeration children(Position p)

Query methods

boolean isInternal(Position p)boolean isExternal(Position p)boolean isRoot(Position p)

Update method

Object replace (Position p Object o)void swap(Position v Position w)

Trees ADT methods

The Trees Abstract Data Type

A Tree Interface in Java

PositionalContainerltltinterfacegtgt

SimpleTreeltltinterfacegtgt

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()bull Object replace (Position p Object o)bull void swap(Position v Position w)

bull Position root()bull Position parent(Position p)bull Enumeration children(Position p)bull boolean isInternal(Position p)bull boolean isExternal(Position p)bull boolean isRoot(Position p)

Basic Algorithms on Trees

Depth and Height

bull The depth of a node v is the number of ancestors of v excluding v itself

bull If v is the root then the depth of v is 0bull Otherwise the depth of v is one plus the depth of the parent v

Basic Algorithms on Trees

Depth and Height

public int depth(SimpleTree TPosition v)

if(TisRoot(v))

return(0)

else

return(1+depth(Tparent(v)))

Basic Algorithms on Trees

Depth and Height

bull The height of an entire tree T is defined to be the height of the root of T Another way to view the height of a tree T is the maximum depth of a node of T which is achieved at one or more external nodes

Basic Algorithms on Trees

Depth and Height

public int height1(SimpleTree T)

int h

Enumberation nodes_of_T=Tpositions()

while(nodes_of_ThasMoreElements())

Position v=(Position)nodes_of_TnextElement()

if(TisExternal(v))

h=Mathmax(hdepth(Tv))

return h

The Trees Abstract Data Type

Example

Height Algorithm 1

Basic Algorithms on Trees

Depth and Height

public int height2(SimpleTree T Position v)if(TisExternal(v))return 0elseint h=0Enumeration children_of_v= Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()h=Mathmax(hTheight2(w))return 1+h

Preorder Traversal

Definition

bull A traversal of a tree T is a systematic way of accessing or ldquovisitingrdquo all the nodes of T

bull In a preorder traversal of a tree T the root of T is visited first and then the subtree rooted at its children are traversed recursively

Preorder TraversalAlgorithm

Algorithm preorder(Tv)Visit(v)for each child w of v dopreorder(Tw)

Preorder TraversalPreorder Print

public void preorderPrint(SimpleTree TPosition v)Systemoutprintln(Telement(v))Enumeration children_of_v=Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()preorderPrint(Tw)

Postorder TraversalDefinition

bull Recursively traverses the subtrees rooted at the children of the root first and then visits the root

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29

Trees

bull The main terminology for tree data structures comes from family trees with the term ldquoparentrdquo ldquochildrdquo ldquoancestorrdquo and ldquodescendentrdquo

Introduction

Trees

Example Prophets tree [1]

سيدنا على صلي اللهممحمد

[1] httpimg246imageshackusimg24635828618qz1png

The Trees Abstract Data Type

A tree is an abstract data type that stores elements hierarchally With the exception of the top element each element in a tree has a parent element and zero or more children

A tree is usually visualized by placing elements inside ovals and rectangles and by drawing the connections between parents and children with straight lines

Definition

root

The Trees Abstract Data Type

The tree T is a set of nodes storing elements in a parent-child relationship with the following properties

bull T has a distinguished node r called the root of T that has no parent

bull Each node v of T distinct from r has a parent node ubull For every node v of T the root r is ancestor of vbull If a node u is the parent of node v then we say that v is a child of ubull Two nodes that are children of the same parent are siblingsbull A node is external or leave if it has no childrenbull A node is internal if it has one or more children

Terminology and Basic properties

The Trees Abstract Data Type

Terminology and Basic properties

root

Parent of I J K

children of F

The Trees Abstract Data Type

Terminology and Basic properties

Descendant of B A

Descendants of F B and A

The Trees Abstract Data Type

Terminology and Basic properties

Ancestor of all other nodes

Ancestor of I J K

The Trees Abstract Data Type

A tree T is ordered if there is a linear ordering defined for the children of each node in such a way we can identify the children of a node as being the first second third and so on Ordered trees typically indicate the linear relationship existing between siblings by listing them in a sequence or enumeration in the correct order

Terminology and Basic Properties

The Trees Abstract Data Type

Terminology and Basic Properties

Book

Preface Part A Part B References

P1 P2 CH 1 Ch2 Ch3 Ch4 Ch5 P1

The Trees Abstract Data Type

A binary tree is a tree with the following propertiesbull Each internal node has at most two children (exactly two for properbinary trees)bull The children of a node are an ordered pairbull We call the children of an internal node left child and right childbull Alternative recursive definition bull A binary tree is either1048708 a tree consisting of a single node or1048708 a tree whose root has an ordered pair of children each of which is a

binary tree

Terminology and Basic Properties

The Trees Abstract Data Type

Example of a binary tree

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applicationsDecision trees

Binary tree associated with a decision process1048708 internal nodes questions with yesno answer1048708 external nodes decisionsExample dining

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applications

Binary tree associated with an arithmetic expression1048708 internal nodes operators1048708 external nodes operandsExample arithmetic expression tree for theexpression (2 times (a minus 1) + (3 times b))

Terminology and Basic Properties

The Trees Abstract Data Type

Generic methods

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()

Accessor methods

Position root()Position parent(Position p)Enumeration children(Position p)

Query methods

boolean isInternal(Position p)boolean isExternal(Position p)boolean isRoot(Position p)

Update method

Object replace (Position p Object o)void swap(Position v Position w)

Trees ADT methods

The Trees Abstract Data Type

A Tree Interface in Java

PositionalContainerltltinterfacegtgt

SimpleTreeltltinterfacegtgt

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()bull Object replace (Position p Object o)bull void swap(Position v Position w)

bull Position root()bull Position parent(Position p)bull Enumeration children(Position p)bull boolean isInternal(Position p)bull boolean isExternal(Position p)bull boolean isRoot(Position p)

Basic Algorithms on Trees

Depth and Height

bull The depth of a node v is the number of ancestors of v excluding v itself

bull If v is the root then the depth of v is 0bull Otherwise the depth of v is one plus the depth of the parent v

Basic Algorithms on Trees

Depth and Height

public int depth(SimpleTree TPosition v)

if(TisRoot(v))

return(0)

else

return(1+depth(Tparent(v)))

Basic Algorithms on Trees

Depth and Height

bull The height of an entire tree T is defined to be the height of the root of T Another way to view the height of a tree T is the maximum depth of a node of T which is achieved at one or more external nodes

Basic Algorithms on Trees

Depth and Height

public int height1(SimpleTree T)

int h

Enumberation nodes_of_T=Tpositions()

while(nodes_of_ThasMoreElements())

Position v=(Position)nodes_of_TnextElement()

if(TisExternal(v))

h=Mathmax(hdepth(Tv))

return h

The Trees Abstract Data Type

Example

Height Algorithm 1

Basic Algorithms on Trees

Depth and Height

public int height2(SimpleTree T Position v)if(TisExternal(v))return 0elseint h=0Enumeration children_of_v= Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()h=Mathmax(hTheight2(w))return 1+h

Preorder Traversal

Definition

bull A traversal of a tree T is a systematic way of accessing or ldquovisitingrdquo all the nodes of T

bull In a preorder traversal of a tree T the root of T is visited first and then the subtree rooted at its children are traversed recursively

Preorder TraversalAlgorithm

Algorithm preorder(Tv)Visit(v)for each child w of v dopreorder(Tw)

Preorder TraversalPreorder Print

public void preorderPrint(SimpleTree TPosition v)Systemoutprintln(Telement(v))Enumeration children_of_v=Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()preorderPrint(Tw)

Postorder TraversalDefinition

bull Recursively traverses the subtrees rooted at the children of the root first and then visits the root

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29

Trees

Example Prophets tree [1]

سيدنا على صلي اللهممحمد

[1] httpimg246imageshackusimg24635828618qz1png

The Trees Abstract Data Type

A tree is an abstract data type that stores elements hierarchally With the exception of the top element each element in a tree has a parent element and zero or more children

A tree is usually visualized by placing elements inside ovals and rectangles and by drawing the connections between parents and children with straight lines

Definition

root

The Trees Abstract Data Type

The tree T is a set of nodes storing elements in a parent-child relationship with the following properties

bull T has a distinguished node r called the root of T that has no parent

bull Each node v of T distinct from r has a parent node ubull For every node v of T the root r is ancestor of vbull If a node u is the parent of node v then we say that v is a child of ubull Two nodes that are children of the same parent are siblingsbull A node is external or leave if it has no childrenbull A node is internal if it has one or more children

Terminology and Basic properties

The Trees Abstract Data Type

Terminology and Basic properties

root

Parent of I J K

children of F

The Trees Abstract Data Type

Terminology and Basic properties

Descendant of B A

Descendants of F B and A

The Trees Abstract Data Type

Terminology and Basic properties

Ancestor of all other nodes

Ancestor of I J K

The Trees Abstract Data Type

A tree T is ordered if there is a linear ordering defined for the children of each node in such a way we can identify the children of a node as being the first second third and so on Ordered trees typically indicate the linear relationship existing between siblings by listing them in a sequence or enumeration in the correct order

Terminology and Basic Properties

The Trees Abstract Data Type

Terminology and Basic Properties

Book

Preface Part A Part B References

P1 P2 CH 1 Ch2 Ch3 Ch4 Ch5 P1

The Trees Abstract Data Type

A binary tree is a tree with the following propertiesbull Each internal node has at most two children (exactly two for properbinary trees)bull The children of a node are an ordered pairbull We call the children of an internal node left child and right childbull Alternative recursive definition bull A binary tree is either1048708 a tree consisting of a single node or1048708 a tree whose root has an ordered pair of children each of which is a

binary tree

Terminology and Basic Properties

The Trees Abstract Data Type

Example of a binary tree

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applicationsDecision trees

Binary tree associated with a decision process1048708 internal nodes questions with yesno answer1048708 external nodes decisionsExample dining

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applications

Binary tree associated with an arithmetic expression1048708 internal nodes operators1048708 external nodes operandsExample arithmetic expression tree for theexpression (2 times (a minus 1) + (3 times b))

Terminology and Basic Properties

The Trees Abstract Data Type

Generic methods

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()

Accessor methods

Position root()Position parent(Position p)Enumeration children(Position p)

Query methods

boolean isInternal(Position p)boolean isExternal(Position p)boolean isRoot(Position p)

Update method

Object replace (Position p Object o)void swap(Position v Position w)

Trees ADT methods

The Trees Abstract Data Type

A Tree Interface in Java

PositionalContainerltltinterfacegtgt

SimpleTreeltltinterfacegtgt

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()bull Object replace (Position p Object o)bull void swap(Position v Position w)

bull Position root()bull Position parent(Position p)bull Enumeration children(Position p)bull boolean isInternal(Position p)bull boolean isExternal(Position p)bull boolean isRoot(Position p)

Basic Algorithms on Trees

Depth and Height

bull The depth of a node v is the number of ancestors of v excluding v itself

bull If v is the root then the depth of v is 0bull Otherwise the depth of v is one plus the depth of the parent v

Basic Algorithms on Trees

Depth and Height

public int depth(SimpleTree TPosition v)

if(TisRoot(v))

return(0)

else

return(1+depth(Tparent(v)))

Basic Algorithms on Trees

Depth and Height

bull The height of an entire tree T is defined to be the height of the root of T Another way to view the height of a tree T is the maximum depth of a node of T which is achieved at one or more external nodes

Basic Algorithms on Trees

Depth and Height

public int height1(SimpleTree T)

int h

Enumberation nodes_of_T=Tpositions()

while(nodes_of_ThasMoreElements())

Position v=(Position)nodes_of_TnextElement()

if(TisExternal(v))

h=Mathmax(hdepth(Tv))

return h

The Trees Abstract Data Type

Example

Height Algorithm 1

Basic Algorithms on Trees

Depth and Height

public int height2(SimpleTree T Position v)if(TisExternal(v))return 0elseint h=0Enumeration children_of_v= Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()h=Mathmax(hTheight2(w))return 1+h

Preorder Traversal

Definition

bull A traversal of a tree T is a systematic way of accessing or ldquovisitingrdquo all the nodes of T

bull In a preorder traversal of a tree T the root of T is visited first and then the subtree rooted at its children are traversed recursively

Preorder TraversalAlgorithm

Algorithm preorder(Tv)Visit(v)for each child w of v dopreorder(Tw)

Preorder TraversalPreorder Print

public void preorderPrint(SimpleTree TPosition v)Systemoutprintln(Telement(v))Enumeration children_of_v=Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()preorderPrint(Tw)

Postorder TraversalDefinition

bull Recursively traverses the subtrees rooted at the children of the root first and then visits the root

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29

The Trees Abstract Data Type

A tree is an abstract data type that stores elements hierarchally With the exception of the top element each element in a tree has a parent element and zero or more children

A tree is usually visualized by placing elements inside ovals and rectangles and by drawing the connections between parents and children with straight lines

Definition

root

The Trees Abstract Data Type

The tree T is a set of nodes storing elements in a parent-child relationship with the following properties

bull T has a distinguished node r called the root of T that has no parent

bull Each node v of T distinct from r has a parent node ubull For every node v of T the root r is ancestor of vbull If a node u is the parent of node v then we say that v is a child of ubull Two nodes that are children of the same parent are siblingsbull A node is external or leave if it has no childrenbull A node is internal if it has one or more children

Terminology and Basic properties

The Trees Abstract Data Type

Terminology and Basic properties

root

Parent of I J K

children of F

The Trees Abstract Data Type

Terminology and Basic properties

Descendant of B A

Descendants of F B and A

The Trees Abstract Data Type

Terminology and Basic properties

Ancestor of all other nodes

Ancestor of I J K

The Trees Abstract Data Type

A tree T is ordered if there is a linear ordering defined for the children of each node in such a way we can identify the children of a node as being the first second third and so on Ordered trees typically indicate the linear relationship existing between siblings by listing them in a sequence or enumeration in the correct order

Terminology and Basic Properties

The Trees Abstract Data Type

Terminology and Basic Properties

Book

Preface Part A Part B References

P1 P2 CH 1 Ch2 Ch3 Ch4 Ch5 P1

The Trees Abstract Data Type

A binary tree is a tree with the following propertiesbull Each internal node has at most two children (exactly two for properbinary trees)bull The children of a node are an ordered pairbull We call the children of an internal node left child and right childbull Alternative recursive definition bull A binary tree is either1048708 a tree consisting of a single node or1048708 a tree whose root has an ordered pair of children each of which is a

binary tree

Terminology and Basic Properties

The Trees Abstract Data Type

Example of a binary tree

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applicationsDecision trees

Binary tree associated with a decision process1048708 internal nodes questions with yesno answer1048708 external nodes decisionsExample dining

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applications

Binary tree associated with an arithmetic expression1048708 internal nodes operators1048708 external nodes operandsExample arithmetic expression tree for theexpression (2 times (a minus 1) + (3 times b))

Terminology and Basic Properties

The Trees Abstract Data Type

Generic methods

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()

Accessor methods

Position root()Position parent(Position p)Enumeration children(Position p)

Query methods

boolean isInternal(Position p)boolean isExternal(Position p)boolean isRoot(Position p)

Update method

Object replace (Position p Object o)void swap(Position v Position w)

Trees ADT methods

The Trees Abstract Data Type

A Tree Interface in Java

PositionalContainerltltinterfacegtgt

SimpleTreeltltinterfacegtgt

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()bull Object replace (Position p Object o)bull void swap(Position v Position w)

bull Position root()bull Position parent(Position p)bull Enumeration children(Position p)bull boolean isInternal(Position p)bull boolean isExternal(Position p)bull boolean isRoot(Position p)

Basic Algorithms on Trees

Depth and Height

bull The depth of a node v is the number of ancestors of v excluding v itself

bull If v is the root then the depth of v is 0bull Otherwise the depth of v is one plus the depth of the parent v

Basic Algorithms on Trees

Depth and Height

public int depth(SimpleTree TPosition v)

if(TisRoot(v))

return(0)

else

return(1+depth(Tparent(v)))

Basic Algorithms on Trees

Depth and Height

bull The height of an entire tree T is defined to be the height of the root of T Another way to view the height of a tree T is the maximum depth of a node of T which is achieved at one or more external nodes

Basic Algorithms on Trees

Depth and Height

public int height1(SimpleTree T)

int h

Enumberation nodes_of_T=Tpositions()

while(nodes_of_ThasMoreElements())

Position v=(Position)nodes_of_TnextElement()

if(TisExternal(v))

h=Mathmax(hdepth(Tv))

return h

The Trees Abstract Data Type

Example

Height Algorithm 1

Basic Algorithms on Trees

Depth and Height

public int height2(SimpleTree T Position v)if(TisExternal(v))return 0elseint h=0Enumeration children_of_v= Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()h=Mathmax(hTheight2(w))return 1+h

Preorder Traversal

Definition

bull A traversal of a tree T is a systematic way of accessing or ldquovisitingrdquo all the nodes of T

bull In a preorder traversal of a tree T the root of T is visited first and then the subtree rooted at its children are traversed recursively

Preorder TraversalAlgorithm

Algorithm preorder(Tv)Visit(v)for each child w of v dopreorder(Tw)

Preorder TraversalPreorder Print

public void preorderPrint(SimpleTree TPosition v)Systemoutprintln(Telement(v))Enumeration children_of_v=Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()preorderPrint(Tw)

Postorder TraversalDefinition

bull Recursively traverses the subtrees rooted at the children of the root first and then visits the root

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29

The Trees Abstract Data Type

The tree T is a set of nodes storing elements in a parent-child relationship with the following properties

bull T has a distinguished node r called the root of T that has no parent

bull Each node v of T distinct from r has a parent node ubull For every node v of T the root r is ancestor of vbull If a node u is the parent of node v then we say that v is a child of ubull Two nodes that are children of the same parent are siblingsbull A node is external or leave if it has no childrenbull A node is internal if it has one or more children

Terminology and Basic properties

The Trees Abstract Data Type

Terminology and Basic properties

root

Parent of I J K

children of F

The Trees Abstract Data Type

Terminology and Basic properties

Descendant of B A

Descendants of F B and A

The Trees Abstract Data Type

Terminology and Basic properties

Ancestor of all other nodes

Ancestor of I J K

The Trees Abstract Data Type

A tree T is ordered if there is a linear ordering defined for the children of each node in such a way we can identify the children of a node as being the first second third and so on Ordered trees typically indicate the linear relationship existing between siblings by listing them in a sequence or enumeration in the correct order

Terminology and Basic Properties

The Trees Abstract Data Type

Terminology and Basic Properties

Book

Preface Part A Part B References

P1 P2 CH 1 Ch2 Ch3 Ch4 Ch5 P1

The Trees Abstract Data Type

A binary tree is a tree with the following propertiesbull Each internal node has at most two children (exactly two for properbinary trees)bull The children of a node are an ordered pairbull We call the children of an internal node left child and right childbull Alternative recursive definition bull A binary tree is either1048708 a tree consisting of a single node or1048708 a tree whose root has an ordered pair of children each of which is a

binary tree

Terminology and Basic Properties

The Trees Abstract Data Type

Example of a binary tree

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applicationsDecision trees

Binary tree associated with a decision process1048708 internal nodes questions with yesno answer1048708 external nodes decisionsExample dining

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applications

Binary tree associated with an arithmetic expression1048708 internal nodes operators1048708 external nodes operandsExample arithmetic expression tree for theexpression (2 times (a minus 1) + (3 times b))

Terminology and Basic Properties

The Trees Abstract Data Type

Generic methods

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()

Accessor methods

Position root()Position parent(Position p)Enumeration children(Position p)

Query methods

boolean isInternal(Position p)boolean isExternal(Position p)boolean isRoot(Position p)

Update method

Object replace (Position p Object o)void swap(Position v Position w)

Trees ADT methods

The Trees Abstract Data Type

A Tree Interface in Java

PositionalContainerltltinterfacegtgt

SimpleTreeltltinterfacegtgt

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()bull Object replace (Position p Object o)bull void swap(Position v Position w)

bull Position root()bull Position parent(Position p)bull Enumeration children(Position p)bull boolean isInternal(Position p)bull boolean isExternal(Position p)bull boolean isRoot(Position p)

Basic Algorithms on Trees

Depth and Height

bull The depth of a node v is the number of ancestors of v excluding v itself

bull If v is the root then the depth of v is 0bull Otherwise the depth of v is one plus the depth of the parent v

Basic Algorithms on Trees

Depth and Height

public int depth(SimpleTree TPosition v)

if(TisRoot(v))

return(0)

else

return(1+depth(Tparent(v)))

Basic Algorithms on Trees

Depth and Height

bull The height of an entire tree T is defined to be the height of the root of T Another way to view the height of a tree T is the maximum depth of a node of T which is achieved at one or more external nodes

Basic Algorithms on Trees

Depth and Height

public int height1(SimpleTree T)

int h

Enumberation nodes_of_T=Tpositions()

while(nodes_of_ThasMoreElements())

Position v=(Position)nodes_of_TnextElement()

if(TisExternal(v))

h=Mathmax(hdepth(Tv))

return h

The Trees Abstract Data Type

Example

Height Algorithm 1

Basic Algorithms on Trees

Depth and Height

public int height2(SimpleTree T Position v)if(TisExternal(v))return 0elseint h=0Enumeration children_of_v= Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()h=Mathmax(hTheight2(w))return 1+h

Preorder Traversal

Definition

bull A traversal of a tree T is a systematic way of accessing or ldquovisitingrdquo all the nodes of T

bull In a preorder traversal of a tree T the root of T is visited first and then the subtree rooted at its children are traversed recursively

Preorder TraversalAlgorithm

Algorithm preorder(Tv)Visit(v)for each child w of v dopreorder(Tw)

Preorder TraversalPreorder Print

public void preorderPrint(SimpleTree TPosition v)Systemoutprintln(Telement(v))Enumeration children_of_v=Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()preorderPrint(Tw)

Postorder TraversalDefinition

bull Recursively traverses the subtrees rooted at the children of the root first and then visits the root

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29

The Trees Abstract Data Type

Terminology and Basic properties

root

Parent of I J K

children of F

The Trees Abstract Data Type

Terminology and Basic properties

Descendant of B A

Descendants of F B and A

The Trees Abstract Data Type

Terminology and Basic properties

Ancestor of all other nodes

Ancestor of I J K

The Trees Abstract Data Type

A tree T is ordered if there is a linear ordering defined for the children of each node in such a way we can identify the children of a node as being the first second third and so on Ordered trees typically indicate the linear relationship existing between siblings by listing them in a sequence or enumeration in the correct order

Terminology and Basic Properties

The Trees Abstract Data Type

Terminology and Basic Properties

Book

Preface Part A Part B References

P1 P2 CH 1 Ch2 Ch3 Ch4 Ch5 P1

The Trees Abstract Data Type

A binary tree is a tree with the following propertiesbull Each internal node has at most two children (exactly two for properbinary trees)bull The children of a node are an ordered pairbull We call the children of an internal node left child and right childbull Alternative recursive definition bull A binary tree is either1048708 a tree consisting of a single node or1048708 a tree whose root has an ordered pair of children each of which is a

binary tree

Terminology and Basic Properties

The Trees Abstract Data Type

Example of a binary tree

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applicationsDecision trees

Binary tree associated with a decision process1048708 internal nodes questions with yesno answer1048708 external nodes decisionsExample dining

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applications

Binary tree associated with an arithmetic expression1048708 internal nodes operators1048708 external nodes operandsExample arithmetic expression tree for theexpression (2 times (a minus 1) + (3 times b))

Terminology and Basic Properties

The Trees Abstract Data Type

Generic methods

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()

Accessor methods

Position root()Position parent(Position p)Enumeration children(Position p)

Query methods

boolean isInternal(Position p)boolean isExternal(Position p)boolean isRoot(Position p)

Update method

Object replace (Position p Object o)void swap(Position v Position w)

Trees ADT methods

The Trees Abstract Data Type

A Tree Interface in Java

PositionalContainerltltinterfacegtgt

SimpleTreeltltinterfacegtgt

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()bull Object replace (Position p Object o)bull void swap(Position v Position w)

bull Position root()bull Position parent(Position p)bull Enumeration children(Position p)bull boolean isInternal(Position p)bull boolean isExternal(Position p)bull boolean isRoot(Position p)

Basic Algorithms on Trees

Depth and Height

bull The depth of a node v is the number of ancestors of v excluding v itself

bull If v is the root then the depth of v is 0bull Otherwise the depth of v is one plus the depth of the parent v

Basic Algorithms on Trees

Depth and Height

public int depth(SimpleTree TPosition v)

if(TisRoot(v))

return(0)

else

return(1+depth(Tparent(v)))

Basic Algorithms on Trees

Depth and Height

bull The height of an entire tree T is defined to be the height of the root of T Another way to view the height of a tree T is the maximum depth of a node of T which is achieved at one or more external nodes

Basic Algorithms on Trees

Depth and Height

public int height1(SimpleTree T)

int h

Enumberation nodes_of_T=Tpositions()

while(nodes_of_ThasMoreElements())

Position v=(Position)nodes_of_TnextElement()

if(TisExternal(v))

h=Mathmax(hdepth(Tv))

return h

The Trees Abstract Data Type

Example

Height Algorithm 1

Basic Algorithms on Trees

Depth and Height

public int height2(SimpleTree T Position v)if(TisExternal(v))return 0elseint h=0Enumeration children_of_v= Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()h=Mathmax(hTheight2(w))return 1+h

Preorder Traversal

Definition

bull A traversal of a tree T is a systematic way of accessing or ldquovisitingrdquo all the nodes of T

bull In a preorder traversal of a tree T the root of T is visited first and then the subtree rooted at its children are traversed recursively

Preorder TraversalAlgorithm

Algorithm preorder(Tv)Visit(v)for each child w of v dopreorder(Tw)

Preorder TraversalPreorder Print

public void preorderPrint(SimpleTree TPosition v)Systemoutprintln(Telement(v))Enumeration children_of_v=Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()preorderPrint(Tw)

Postorder TraversalDefinition

bull Recursively traverses the subtrees rooted at the children of the root first and then visits the root

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29

The Trees Abstract Data Type

Terminology and Basic properties

Descendant of B A

Descendants of F B and A

The Trees Abstract Data Type

Terminology and Basic properties

Ancestor of all other nodes

Ancestor of I J K

The Trees Abstract Data Type

A tree T is ordered if there is a linear ordering defined for the children of each node in such a way we can identify the children of a node as being the first second third and so on Ordered trees typically indicate the linear relationship existing between siblings by listing them in a sequence or enumeration in the correct order

Terminology and Basic Properties

The Trees Abstract Data Type

Terminology and Basic Properties

Book

Preface Part A Part B References

P1 P2 CH 1 Ch2 Ch3 Ch4 Ch5 P1

The Trees Abstract Data Type

A binary tree is a tree with the following propertiesbull Each internal node has at most two children (exactly two for properbinary trees)bull The children of a node are an ordered pairbull We call the children of an internal node left child and right childbull Alternative recursive definition bull A binary tree is either1048708 a tree consisting of a single node or1048708 a tree whose root has an ordered pair of children each of which is a

binary tree

Terminology and Basic Properties

The Trees Abstract Data Type

Example of a binary tree

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applicationsDecision trees

Binary tree associated with a decision process1048708 internal nodes questions with yesno answer1048708 external nodes decisionsExample dining

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applications

Binary tree associated with an arithmetic expression1048708 internal nodes operators1048708 external nodes operandsExample arithmetic expression tree for theexpression (2 times (a minus 1) + (3 times b))

Terminology and Basic Properties

The Trees Abstract Data Type

Generic methods

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()

Accessor methods

Position root()Position parent(Position p)Enumeration children(Position p)

Query methods

boolean isInternal(Position p)boolean isExternal(Position p)boolean isRoot(Position p)

Update method

Object replace (Position p Object o)void swap(Position v Position w)

Trees ADT methods

The Trees Abstract Data Type

A Tree Interface in Java

PositionalContainerltltinterfacegtgt

SimpleTreeltltinterfacegtgt

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()bull Object replace (Position p Object o)bull void swap(Position v Position w)

bull Position root()bull Position parent(Position p)bull Enumeration children(Position p)bull boolean isInternal(Position p)bull boolean isExternal(Position p)bull boolean isRoot(Position p)

Basic Algorithms on Trees

Depth and Height

bull The depth of a node v is the number of ancestors of v excluding v itself

bull If v is the root then the depth of v is 0bull Otherwise the depth of v is one plus the depth of the parent v

Basic Algorithms on Trees

Depth and Height

public int depth(SimpleTree TPosition v)

if(TisRoot(v))

return(0)

else

return(1+depth(Tparent(v)))

Basic Algorithms on Trees

Depth and Height

bull The height of an entire tree T is defined to be the height of the root of T Another way to view the height of a tree T is the maximum depth of a node of T which is achieved at one or more external nodes

Basic Algorithms on Trees

Depth and Height

public int height1(SimpleTree T)

int h

Enumberation nodes_of_T=Tpositions()

while(nodes_of_ThasMoreElements())

Position v=(Position)nodes_of_TnextElement()

if(TisExternal(v))

h=Mathmax(hdepth(Tv))

return h

The Trees Abstract Data Type

Example

Height Algorithm 1

Basic Algorithms on Trees

Depth and Height

public int height2(SimpleTree T Position v)if(TisExternal(v))return 0elseint h=0Enumeration children_of_v= Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()h=Mathmax(hTheight2(w))return 1+h

Preorder Traversal

Definition

bull A traversal of a tree T is a systematic way of accessing or ldquovisitingrdquo all the nodes of T

bull In a preorder traversal of a tree T the root of T is visited first and then the subtree rooted at its children are traversed recursively

Preorder TraversalAlgorithm

Algorithm preorder(Tv)Visit(v)for each child w of v dopreorder(Tw)

Preorder TraversalPreorder Print

public void preorderPrint(SimpleTree TPosition v)Systemoutprintln(Telement(v))Enumeration children_of_v=Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()preorderPrint(Tw)

Postorder TraversalDefinition

bull Recursively traverses the subtrees rooted at the children of the root first and then visits the root

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29

The Trees Abstract Data Type

Terminology and Basic properties

Ancestor of all other nodes

Ancestor of I J K

The Trees Abstract Data Type

A tree T is ordered if there is a linear ordering defined for the children of each node in such a way we can identify the children of a node as being the first second third and so on Ordered trees typically indicate the linear relationship existing between siblings by listing them in a sequence or enumeration in the correct order

Terminology and Basic Properties

The Trees Abstract Data Type

Terminology and Basic Properties

Book

Preface Part A Part B References

P1 P2 CH 1 Ch2 Ch3 Ch4 Ch5 P1

The Trees Abstract Data Type

A binary tree is a tree with the following propertiesbull Each internal node has at most two children (exactly two for properbinary trees)bull The children of a node are an ordered pairbull We call the children of an internal node left child and right childbull Alternative recursive definition bull A binary tree is either1048708 a tree consisting of a single node or1048708 a tree whose root has an ordered pair of children each of which is a

binary tree

Terminology and Basic Properties

The Trees Abstract Data Type

Example of a binary tree

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applicationsDecision trees

Binary tree associated with a decision process1048708 internal nodes questions with yesno answer1048708 external nodes decisionsExample dining

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applications

Binary tree associated with an arithmetic expression1048708 internal nodes operators1048708 external nodes operandsExample arithmetic expression tree for theexpression (2 times (a minus 1) + (3 times b))

Terminology and Basic Properties

The Trees Abstract Data Type

Generic methods

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()

Accessor methods

Position root()Position parent(Position p)Enumeration children(Position p)

Query methods

boolean isInternal(Position p)boolean isExternal(Position p)boolean isRoot(Position p)

Update method

Object replace (Position p Object o)void swap(Position v Position w)

Trees ADT methods

The Trees Abstract Data Type

A Tree Interface in Java

PositionalContainerltltinterfacegtgt

SimpleTreeltltinterfacegtgt

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()bull Object replace (Position p Object o)bull void swap(Position v Position w)

bull Position root()bull Position parent(Position p)bull Enumeration children(Position p)bull boolean isInternal(Position p)bull boolean isExternal(Position p)bull boolean isRoot(Position p)

Basic Algorithms on Trees

Depth and Height

bull The depth of a node v is the number of ancestors of v excluding v itself

bull If v is the root then the depth of v is 0bull Otherwise the depth of v is one plus the depth of the parent v

Basic Algorithms on Trees

Depth and Height

public int depth(SimpleTree TPosition v)

if(TisRoot(v))

return(0)

else

return(1+depth(Tparent(v)))

Basic Algorithms on Trees

Depth and Height

bull The height of an entire tree T is defined to be the height of the root of T Another way to view the height of a tree T is the maximum depth of a node of T which is achieved at one or more external nodes

Basic Algorithms on Trees

Depth and Height

public int height1(SimpleTree T)

int h

Enumberation nodes_of_T=Tpositions()

while(nodes_of_ThasMoreElements())

Position v=(Position)nodes_of_TnextElement()

if(TisExternal(v))

h=Mathmax(hdepth(Tv))

return h

The Trees Abstract Data Type

Example

Height Algorithm 1

Basic Algorithms on Trees

Depth and Height

public int height2(SimpleTree T Position v)if(TisExternal(v))return 0elseint h=0Enumeration children_of_v= Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()h=Mathmax(hTheight2(w))return 1+h

Preorder Traversal

Definition

bull A traversal of a tree T is a systematic way of accessing or ldquovisitingrdquo all the nodes of T

bull In a preorder traversal of a tree T the root of T is visited first and then the subtree rooted at its children are traversed recursively

Preorder TraversalAlgorithm

Algorithm preorder(Tv)Visit(v)for each child w of v dopreorder(Tw)

Preorder TraversalPreorder Print

public void preorderPrint(SimpleTree TPosition v)Systemoutprintln(Telement(v))Enumeration children_of_v=Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()preorderPrint(Tw)

Postorder TraversalDefinition

bull Recursively traverses the subtrees rooted at the children of the root first and then visits the root

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29

The Trees Abstract Data Type

A tree T is ordered if there is a linear ordering defined for the children of each node in such a way we can identify the children of a node as being the first second third and so on Ordered trees typically indicate the linear relationship existing between siblings by listing them in a sequence or enumeration in the correct order

Terminology and Basic Properties

The Trees Abstract Data Type

Terminology and Basic Properties

Book

Preface Part A Part B References

P1 P2 CH 1 Ch2 Ch3 Ch4 Ch5 P1

The Trees Abstract Data Type

A binary tree is a tree with the following propertiesbull Each internal node has at most two children (exactly two for properbinary trees)bull The children of a node are an ordered pairbull We call the children of an internal node left child and right childbull Alternative recursive definition bull A binary tree is either1048708 a tree consisting of a single node or1048708 a tree whose root has an ordered pair of children each of which is a

binary tree

Terminology and Basic Properties

The Trees Abstract Data Type

Example of a binary tree

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applicationsDecision trees

Binary tree associated with a decision process1048708 internal nodes questions with yesno answer1048708 external nodes decisionsExample dining

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applications

Binary tree associated with an arithmetic expression1048708 internal nodes operators1048708 external nodes operandsExample arithmetic expression tree for theexpression (2 times (a minus 1) + (3 times b))

Terminology and Basic Properties

The Trees Abstract Data Type

Generic methods

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()

Accessor methods

Position root()Position parent(Position p)Enumeration children(Position p)

Query methods

boolean isInternal(Position p)boolean isExternal(Position p)boolean isRoot(Position p)

Update method

Object replace (Position p Object o)void swap(Position v Position w)

Trees ADT methods

The Trees Abstract Data Type

A Tree Interface in Java

PositionalContainerltltinterfacegtgt

SimpleTreeltltinterfacegtgt

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()bull Object replace (Position p Object o)bull void swap(Position v Position w)

bull Position root()bull Position parent(Position p)bull Enumeration children(Position p)bull boolean isInternal(Position p)bull boolean isExternal(Position p)bull boolean isRoot(Position p)

Basic Algorithms on Trees

Depth and Height

bull The depth of a node v is the number of ancestors of v excluding v itself

bull If v is the root then the depth of v is 0bull Otherwise the depth of v is one plus the depth of the parent v

Basic Algorithms on Trees

Depth and Height

public int depth(SimpleTree TPosition v)

if(TisRoot(v))

return(0)

else

return(1+depth(Tparent(v)))

Basic Algorithms on Trees

Depth and Height

bull The height of an entire tree T is defined to be the height of the root of T Another way to view the height of a tree T is the maximum depth of a node of T which is achieved at one or more external nodes

Basic Algorithms on Trees

Depth and Height

public int height1(SimpleTree T)

int h

Enumberation nodes_of_T=Tpositions()

while(nodes_of_ThasMoreElements())

Position v=(Position)nodes_of_TnextElement()

if(TisExternal(v))

h=Mathmax(hdepth(Tv))

return h

The Trees Abstract Data Type

Example

Height Algorithm 1

Basic Algorithms on Trees

Depth and Height

public int height2(SimpleTree T Position v)if(TisExternal(v))return 0elseint h=0Enumeration children_of_v= Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()h=Mathmax(hTheight2(w))return 1+h

Preorder Traversal

Definition

bull A traversal of a tree T is a systematic way of accessing or ldquovisitingrdquo all the nodes of T

bull In a preorder traversal of a tree T the root of T is visited first and then the subtree rooted at its children are traversed recursively

Preorder TraversalAlgorithm

Algorithm preorder(Tv)Visit(v)for each child w of v dopreorder(Tw)

Preorder TraversalPreorder Print

public void preorderPrint(SimpleTree TPosition v)Systemoutprintln(Telement(v))Enumeration children_of_v=Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()preorderPrint(Tw)

Postorder TraversalDefinition

bull Recursively traverses the subtrees rooted at the children of the root first and then visits the root

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29

The Trees Abstract Data Type

Terminology and Basic Properties

Book

Preface Part A Part B References

P1 P2 CH 1 Ch2 Ch3 Ch4 Ch5 P1

The Trees Abstract Data Type

A binary tree is a tree with the following propertiesbull Each internal node has at most two children (exactly two for properbinary trees)bull The children of a node are an ordered pairbull We call the children of an internal node left child and right childbull Alternative recursive definition bull A binary tree is either1048708 a tree consisting of a single node or1048708 a tree whose root has an ordered pair of children each of which is a

binary tree

Terminology and Basic Properties

The Trees Abstract Data Type

Example of a binary tree

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applicationsDecision trees

Binary tree associated with a decision process1048708 internal nodes questions with yesno answer1048708 external nodes decisionsExample dining

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applications

Binary tree associated with an arithmetic expression1048708 internal nodes operators1048708 external nodes operandsExample arithmetic expression tree for theexpression (2 times (a minus 1) + (3 times b))

Terminology and Basic Properties

The Trees Abstract Data Type

Generic methods

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()

Accessor methods

Position root()Position parent(Position p)Enumeration children(Position p)

Query methods

boolean isInternal(Position p)boolean isExternal(Position p)boolean isRoot(Position p)

Update method

Object replace (Position p Object o)void swap(Position v Position w)

Trees ADT methods

The Trees Abstract Data Type

A Tree Interface in Java

PositionalContainerltltinterfacegtgt

SimpleTreeltltinterfacegtgt

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()bull Object replace (Position p Object o)bull void swap(Position v Position w)

bull Position root()bull Position parent(Position p)bull Enumeration children(Position p)bull boolean isInternal(Position p)bull boolean isExternal(Position p)bull boolean isRoot(Position p)

Basic Algorithms on Trees

Depth and Height

bull The depth of a node v is the number of ancestors of v excluding v itself

bull If v is the root then the depth of v is 0bull Otherwise the depth of v is one plus the depth of the parent v

Basic Algorithms on Trees

Depth and Height

public int depth(SimpleTree TPosition v)

if(TisRoot(v))

return(0)

else

return(1+depth(Tparent(v)))

Basic Algorithms on Trees

Depth and Height

bull The height of an entire tree T is defined to be the height of the root of T Another way to view the height of a tree T is the maximum depth of a node of T which is achieved at one or more external nodes

Basic Algorithms on Trees

Depth and Height

public int height1(SimpleTree T)

int h

Enumberation nodes_of_T=Tpositions()

while(nodes_of_ThasMoreElements())

Position v=(Position)nodes_of_TnextElement()

if(TisExternal(v))

h=Mathmax(hdepth(Tv))

return h

The Trees Abstract Data Type

Example

Height Algorithm 1

Basic Algorithms on Trees

Depth and Height

public int height2(SimpleTree T Position v)if(TisExternal(v))return 0elseint h=0Enumeration children_of_v= Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()h=Mathmax(hTheight2(w))return 1+h

Preorder Traversal

Definition

bull A traversal of a tree T is a systematic way of accessing or ldquovisitingrdquo all the nodes of T

bull In a preorder traversal of a tree T the root of T is visited first and then the subtree rooted at its children are traversed recursively

Preorder TraversalAlgorithm

Algorithm preorder(Tv)Visit(v)for each child w of v dopreorder(Tw)

Preorder TraversalPreorder Print

public void preorderPrint(SimpleTree TPosition v)Systemoutprintln(Telement(v))Enumeration children_of_v=Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()preorderPrint(Tw)

Postorder TraversalDefinition

bull Recursively traverses the subtrees rooted at the children of the root first and then visits the root

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29

The Trees Abstract Data Type

A binary tree is a tree with the following propertiesbull Each internal node has at most two children (exactly two for properbinary trees)bull The children of a node are an ordered pairbull We call the children of an internal node left child and right childbull Alternative recursive definition bull A binary tree is either1048708 a tree consisting of a single node or1048708 a tree whose root has an ordered pair of children each of which is a

binary tree

Terminology and Basic Properties

The Trees Abstract Data Type

Example of a binary tree

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applicationsDecision trees

Binary tree associated with a decision process1048708 internal nodes questions with yesno answer1048708 external nodes decisionsExample dining

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applications

Binary tree associated with an arithmetic expression1048708 internal nodes operators1048708 external nodes operandsExample arithmetic expression tree for theexpression (2 times (a minus 1) + (3 times b))

Terminology and Basic Properties

The Trees Abstract Data Type

Generic methods

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()

Accessor methods

Position root()Position parent(Position p)Enumeration children(Position p)

Query methods

boolean isInternal(Position p)boolean isExternal(Position p)boolean isRoot(Position p)

Update method

Object replace (Position p Object o)void swap(Position v Position w)

Trees ADT methods

The Trees Abstract Data Type

A Tree Interface in Java

PositionalContainerltltinterfacegtgt

SimpleTreeltltinterfacegtgt

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()bull Object replace (Position p Object o)bull void swap(Position v Position w)

bull Position root()bull Position parent(Position p)bull Enumeration children(Position p)bull boolean isInternal(Position p)bull boolean isExternal(Position p)bull boolean isRoot(Position p)

Basic Algorithms on Trees

Depth and Height

bull The depth of a node v is the number of ancestors of v excluding v itself

bull If v is the root then the depth of v is 0bull Otherwise the depth of v is one plus the depth of the parent v

Basic Algorithms on Trees

Depth and Height

public int depth(SimpleTree TPosition v)

if(TisRoot(v))

return(0)

else

return(1+depth(Tparent(v)))

Basic Algorithms on Trees

Depth and Height

bull The height of an entire tree T is defined to be the height of the root of T Another way to view the height of a tree T is the maximum depth of a node of T which is achieved at one or more external nodes

Basic Algorithms on Trees

Depth and Height

public int height1(SimpleTree T)

int h

Enumberation nodes_of_T=Tpositions()

while(nodes_of_ThasMoreElements())

Position v=(Position)nodes_of_TnextElement()

if(TisExternal(v))

h=Mathmax(hdepth(Tv))

return h

The Trees Abstract Data Type

Example

Height Algorithm 1

Basic Algorithms on Trees

Depth and Height

public int height2(SimpleTree T Position v)if(TisExternal(v))return 0elseint h=0Enumeration children_of_v= Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()h=Mathmax(hTheight2(w))return 1+h

Preorder Traversal

Definition

bull A traversal of a tree T is a systematic way of accessing or ldquovisitingrdquo all the nodes of T

bull In a preorder traversal of a tree T the root of T is visited first and then the subtree rooted at its children are traversed recursively

Preorder TraversalAlgorithm

Algorithm preorder(Tv)Visit(v)for each child w of v dopreorder(Tw)

Preorder TraversalPreorder Print

public void preorderPrint(SimpleTree TPosition v)Systemoutprintln(Telement(v))Enumeration children_of_v=Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()preorderPrint(Tw)

Postorder TraversalDefinition

bull Recursively traverses the subtrees rooted at the children of the root first and then visits the root

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29

The Trees Abstract Data Type

Example of a binary tree

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applicationsDecision trees

Binary tree associated with a decision process1048708 internal nodes questions with yesno answer1048708 external nodes decisionsExample dining

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applications

Binary tree associated with an arithmetic expression1048708 internal nodes operators1048708 external nodes operandsExample arithmetic expression tree for theexpression (2 times (a minus 1) + (3 times b))

Terminology and Basic Properties

The Trees Abstract Data Type

Generic methods

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()

Accessor methods

Position root()Position parent(Position p)Enumeration children(Position p)

Query methods

boolean isInternal(Position p)boolean isExternal(Position p)boolean isRoot(Position p)

Update method

Object replace (Position p Object o)void swap(Position v Position w)

Trees ADT methods

The Trees Abstract Data Type

A Tree Interface in Java

PositionalContainerltltinterfacegtgt

SimpleTreeltltinterfacegtgt

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()bull Object replace (Position p Object o)bull void swap(Position v Position w)

bull Position root()bull Position parent(Position p)bull Enumeration children(Position p)bull boolean isInternal(Position p)bull boolean isExternal(Position p)bull boolean isRoot(Position p)

Basic Algorithms on Trees

Depth and Height

bull The depth of a node v is the number of ancestors of v excluding v itself

bull If v is the root then the depth of v is 0bull Otherwise the depth of v is one plus the depth of the parent v

Basic Algorithms on Trees

Depth and Height

public int depth(SimpleTree TPosition v)

if(TisRoot(v))

return(0)

else

return(1+depth(Tparent(v)))

Basic Algorithms on Trees

Depth and Height

bull The height of an entire tree T is defined to be the height of the root of T Another way to view the height of a tree T is the maximum depth of a node of T which is achieved at one or more external nodes

Basic Algorithms on Trees

Depth and Height

public int height1(SimpleTree T)

int h

Enumberation nodes_of_T=Tpositions()

while(nodes_of_ThasMoreElements())

Position v=(Position)nodes_of_TnextElement()

if(TisExternal(v))

h=Mathmax(hdepth(Tv))

return h

The Trees Abstract Data Type

Example

Height Algorithm 1

Basic Algorithms on Trees

Depth and Height

public int height2(SimpleTree T Position v)if(TisExternal(v))return 0elseint h=0Enumeration children_of_v= Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()h=Mathmax(hTheight2(w))return 1+h

Preorder Traversal

Definition

bull A traversal of a tree T is a systematic way of accessing or ldquovisitingrdquo all the nodes of T

bull In a preorder traversal of a tree T the root of T is visited first and then the subtree rooted at its children are traversed recursively

Preorder TraversalAlgorithm

Algorithm preorder(Tv)Visit(v)for each child w of v dopreorder(Tw)

Preorder TraversalPreorder Print

public void preorderPrint(SimpleTree TPosition v)Systemoutprintln(Telement(v))Enumeration children_of_v=Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()preorderPrint(Tw)

Postorder TraversalDefinition

bull Recursively traverses the subtrees rooted at the children of the root first and then visits the root

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29

The Trees Abstract Data Type

Binary trees applicationsDecision trees

Binary tree associated with a decision process1048708 internal nodes questions with yesno answer1048708 external nodes decisionsExample dining

Terminology and Basic Properties

The Trees Abstract Data Type

Binary trees applications

Binary tree associated with an arithmetic expression1048708 internal nodes operators1048708 external nodes operandsExample arithmetic expression tree for theexpression (2 times (a minus 1) + (3 times b))

Terminology and Basic Properties

The Trees Abstract Data Type

Generic methods

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()

Accessor methods

Position root()Position parent(Position p)Enumeration children(Position p)

Query methods

boolean isInternal(Position p)boolean isExternal(Position p)boolean isRoot(Position p)

Update method

Object replace (Position p Object o)void swap(Position v Position w)

Trees ADT methods

The Trees Abstract Data Type

A Tree Interface in Java

PositionalContainerltltinterfacegtgt

SimpleTreeltltinterfacegtgt

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()bull Object replace (Position p Object o)bull void swap(Position v Position w)

bull Position root()bull Position parent(Position p)bull Enumeration children(Position p)bull boolean isInternal(Position p)bull boolean isExternal(Position p)bull boolean isRoot(Position p)

Basic Algorithms on Trees

Depth and Height

bull The depth of a node v is the number of ancestors of v excluding v itself

bull If v is the root then the depth of v is 0bull Otherwise the depth of v is one plus the depth of the parent v

Basic Algorithms on Trees

Depth and Height

public int depth(SimpleTree TPosition v)

if(TisRoot(v))

return(0)

else

return(1+depth(Tparent(v)))

Basic Algorithms on Trees

Depth and Height

bull The height of an entire tree T is defined to be the height of the root of T Another way to view the height of a tree T is the maximum depth of a node of T which is achieved at one or more external nodes

Basic Algorithms on Trees

Depth and Height

public int height1(SimpleTree T)

int h

Enumberation nodes_of_T=Tpositions()

while(nodes_of_ThasMoreElements())

Position v=(Position)nodes_of_TnextElement()

if(TisExternal(v))

h=Mathmax(hdepth(Tv))

return h

The Trees Abstract Data Type

Example

Height Algorithm 1

Basic Algorithms on Trees

Depth and Height

public int height2(SimpleTree T Position v)if(TisExternal(v))return 0elseint h=0Enumeration children_of_v= Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()h=Mathmax(hTheight2(w))return 1+h

Preorder Traversal

Definition

bull A traversal of a tree T is a systematic way of accessing or ldquovisitingrdquo all the nodes of T

bull In a preorder traversal of a tree T the root of T is visited first and then the subtree rooted at its children are traversed recursively

Preorder TraversalAlgorithm

Algorithm preorder(Tv)Visit(v)for each child w of v dopreorder(Tw)

Preorder TraversalPreorder Print

public void preorderPrint(SimpleTree TPosition v)Systemoutprintln(Telement(v))Enumeration children_of_v=Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()preorderPrint(Tw)

Postorder TraversalDefinition

bull Recursively traverses the subtrees rooted at the children of the root first and then visits the root

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29

The Trees Abstract Data Type

Binary trees applications

Binary tree associated with an arithmetic expression1048708 internal nodes operators1048708 external nodes operandsExample arithmetic expression tree for theexpression (2 times (a minus 1) + (3 times b))

Terminology and Basic Properties

The Trees Abstract Data Type

Generic methods

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()

Accessor methods

Position root()Position parent(Position p)Enumeration children(Position p)

Query methods

boolean isInternal(Position p)boolean isExternal(Position p)boolean isRoot(Position p)

Update method

Object replace (Position p Object o)void swap(Position v Position w)

Trees ADT methods

The Trees Abstract Data Type

A Tree Interface in Java

PositionalContainerltltinterfacegtgt

SimpleTreeltltinterfacegtgt

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()bull Object replace (Position p Object o)bull void swap(Position v Position w)

bull Position root()bull Position parent(Position p)bull Enumeration children(Position p)bull boolean isInternal(Position p)bull boolean isExternal(Position p)bull boolean isRoot(Position p)

Basic Algorithms on Trees

Depth and Height

bull The depth of a node v is the number of ancestors of v excluding v itself

bull If v is the root then the depth of v is 0bull Otherwise the depth of v is one plus the depth of the parent v

Basic Algorithms on Trees

Depth and Height

public int depth(SimpleTree TPosition v)

if(TisRoot(v))

return(0)

else

return(1+depth(Tparent(v)))

Basic Algorithms on Trees

Depth and Height

bull The height of an entire tree T is defined to be the height of the root of T Another way to view the height of a tree T is the maximum depth of a node of T which is achieved at one or more external nodes

Basic Algorithms on Trees

Depth and Height

public int height1(SimpleTree T)

int h

Enumberation nodes_of_T=Tpositions()

while(nodes_of_ThasMoreElements())

Position v=(Position)nodes_of_TnextElement()

if(TisExternal(v))

h=Mathmax(hdepth(Tv))

return h

The Trees Abstract Data Type

Example

Height Algorithm 1

Basic Algorithms on Trees

Depth and Height

public int height2(SimpleTree T Position v)if(TisExternal(v))return 0elseint h=0Enumeration children_of_v= Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()h=Mathmax(hTheight2(w))return 1+h

Preorder Traversal

Definition

bull A traversal of a tree T is a systematic way of accessing or ldquovisitingrdquo all the nodes of T

bull In a preorder traversal of a tree T the root of T is visited first and then the subtree rooted at its children are traversed recursively

Preorder TraversalAlgorithm

Algorithm preorder(Tv)Visit(v)for each child w of v dopreorder(Tw)

Preorder TraversalPreorder Print

public void preorderPrint(SimpleTree TPosition v)Systemoutprintln(Telement(v))Enumeration children_of_v=Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()preorderPrint(Tw)

Postorder TraversalDefinition

bull Recursively traverses the subtrees rooted at the children of the root first and then visits the root

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29

The Trees Abstract Data Type

Generic methods

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()

Accessor methods

Position root()Position parent(Position p)Enumeration children(Position p)

Query methods

boolean isInternal(Position p)boolean isExternal(Position p)boolean isRoot(Position p)

Update method

Object replace (Position p Object o)void swap(Position v Position w)

Trees ADT methods

The Trees Abstract Data Type

A Tree Interface in Java

PositionalContainerltltinterfacegtgt

SimpleTreeltltinterfacegtgt

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()bull Object replace (Position p Object o)bull void swap(Position v Position w)

bull Position root()bull Position parent(Position p)bull Enumeration children(Position p)bull boolean isInternal(Position p)bull boolean isExternal(Position p)bull boolean isRoot(Position p)

Basic Algorithms on Trees

Depth and Height

bull The depth of a node v is the number of ancestors of v excluding v itself

bull If v is the root then the depth of v is 0bull Otherwise the depth of v is one plus the depth of the parent v

Basic Algorithms on Trees

Depth and Height

public int depth(SimpleTree TPosition v)

if(TisRoot(v))

return(0)

else

return(1+depth(Tparent(v)))

Basic Algorithms on Trees

Depth and Height

bull The height of an entire tree T is defined to be the height of the root of T Another way to view the height of a tree T is the maximum depth of a node of T which is achieved at one or more external nodes

Basic Algorithms on Trees

Depth and Height

public int height1(SimpleTree T)

int h

Enumberation nodes_of_T=Tpositions()

while(nodes_of_ThasMoreElements())

Position v=(Position)nodes_of_TnextElement()

if(TisExternal(v))

h=Mathmax(hdepth(Tv))

return h

The Trees Abstract Data Type

Example

Height Algorithm 1

Basic Algorithms on Trees

Depth and Height

public int height2(SimpleTree T Position v)if(TisExternal(v))return 0elseint h=0Enumeration children_of_v= Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()h=Mathmax(hTheight2(w))return 1+h

Preorder Traversal

Definition

bull A traversal of a tree T is a systematic way of accessing or ldquovisitingrdquo all the nodes of T

bull In a preorder traversal of a tree T the root of T is visited first and then the subtree rooted at its children are traversed recursively

Preorder TraversalAlgorithm

Algorithm preorder(Tv)Visit(v)for each child w of v dopreorder(Tw)

Preorder TraversalPreorder Print

public void preorderPrint(SimpleTree TPosition v)Systemoutprintln(Telement(v))Enumeration children_of_v=Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()preorderPrint(Tw)

Postorder TraversalDefinition

bull Recursively traverses the subtrees rooted at the children of the root first and then visits the root

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29

The Trees Abstract Data Type

A Tree Interface in Java

PositionalContainerltltinterfacegtgt

SimpleTreeltltinterfacegtgt

bull int size()bull boolean isEmpty()bull Enumeration elements()bull Enumeration positions()bull Object replace (Position p Object o)bull void swap(Position v Position w)

bull Position root()bull Position parent(Position p)bull Enumeration children(Position p)bull boolean isInternal(Position p)bull boolean isExternal(Position p)bull boolean isRoot(Position p)

Basic Algorithms on Trees

Depth and Height

bull The depth of a node v is the number of ancestors of v excluding v itself

bull If v is the root then the depth of v is 0bull Otherwise the depth of v is one plus the depth of the parent v

Basic Algorithms on Trees

Depth and Height

public int depth(SimpleTree TPosition v)

if(TisRoot(v))

return(0)

else

return(1+depth(Tparent(v)))

Basic Algorithms on Trees

Depth and Height

bull The height of an entire tree T is defined to be the height of the root of T Another way to view the height of a tree T is the maximum depth of a node of T which is achieved at one or more external nodes

Basic Algorithms on Trees

Depth and Height

public int height1(SimpleTree T)

int h

Enumberation nodes_of_T=Tpositions()

while(nodes_of_ThasMoreElements())

Position v=(Position)nodes_of_TnextElement()

if(TisExternal(v))

h=Mathmax(hdepth(Tv))

return h

The Trees Abstract Data Type

Example

Height Algorithm 1

Basic Algorithms on Trees

Depth and Height

public int height2(SimpleTree T Position v)if(TisExternal(v))return 0elseint h=0Enumeration children_of_v= Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()h=Mathmax(hTheight2(w))return 1+h

Preorder Traversal

Definition

bull A traversal of a tree T is a systematic way of accessing or ldquovisitingrdquo all the nodes of T

bull In a preorder traversal of a tree T the root of T is visited first and then the subtree rooted at its children are traversed recursively

Preorder TraversalAlgorithm

Algorithm preorder(Tv)Visit(v)for each child w of v dopreorder(Tw)

Preorder TraversalPreorder Print

public void preorderPrint(SimpleTree TPosition v)Systemoutprintln(Telement(v))Enumeration children_of_v=Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()preorderPrint(Tw)

Postorder TraversalDefinition

bull Recursively traverses the subtrees rooted at the children of the root first and then visits the root

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29

Basic Algorithms on Trees

Depth and Height

bull The depth of a node v is the number of ancestors of v excluding v itself

bull If v is the root then the depth of v is 0bull Otherwise the depth of v is one plus the depth of the parent v

Basic Algorithms on Trees

Depth and Height

public int depth(SimpleTree TPosition v)

if(TisRoot(v))

return(0)

else

return(1+depth(Tparent(v)))

Basic Algorithms on Trees

Depth and Height

bull The height of an entire tree T is defined to be the height of the root of T Another way to view the height of a tree T is the maximum depth of a node of T which is achieved at one or more external nodes

Basic Algorithms on Trees

Depth and Height

public int height1(SimpleTree T)

int h

Enumberation nodes_of_T=Tpositions()

while(nodes_of_ThasMoreElements())

Position v=(Position)nodes_of_TnextElement()

if(TisExternal(v))

h=Mathmax(hdepth(Tv))

return h

The Trees Abstract Data Type

Example

Height Algorithm 1

Basic Algorithms on Trees

Depth and Height

public int height2(SimpleTree T Position v)if(TisExternal(v))return 0elseint h=0Enumeration children_of_v= Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()h=Mathmax(hTheight2(w))return 1+h

Preorder Traversal

Definition

bull A traversal of a tree T is a systematic way of accessing or ldquovisitingrdquo all the nodes of T

bull In a preorder traversal of a tree T the root of T is visited first and then the subtree rooted at its children are traversed recursively

Preorder TraversalAlgorithm

Algorithm preorder(Tv)Visit(v)for each child w of v dopreorder(Tw)

Preorder TraversalPreorder Print

public void preorderPrint(SimpleTree TPosition v)Systemoutprintln(Telement(v))Enumeration children_of_v=Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()preorderPrint(Tw)

Postorder TraversalDefinition

bull Recursively traverses the subtrees rooted at the children of the root first and then visits the root

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29

Basic Algorithms on Trees

Depth and Height

public int depth(SimpleTree TPosition v)

if(TisRoot(v))

return(0)

else

return(1+depth(Tparent(v)))

Basic Algorithms on Trees

Depth and Height

bull The height of an entire tree T is defined to be the height of the root of T Another way to view the height of a tree T is the maximum depth of a node of T which is achieved at one or more external nodes

Basic Algorithms on Trees

Depth and Height

public int height1(SimpleTree T)

int h

Enumberation nodes_of_T=Tpositions()

while(nodes_of_ThasMoreElements())

Position v=(Position)nodes_of_TnextElement()

if(TisExternal(v))

h=Mathmax(hdepth(Tv))

return h

The Trees Abstract Data Type

Example

Height Algorithm 1

Basic Algorithms on Trees

Depth and Height

public int height2(SimpleTree T Position v)if(TisExternal(v))return 0elseint h=0Enumeration children_of_v= Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()h=Mathmax(hTheight2(w))return 1+h

Preorder Traversal

Definition

bull A traversal of a tree T is a systematic way of accessing or ldquovisitingrdquo all the nodes of T

bull In a preorder traversal of a tree T the root of T is visited first and then the subtree rooted at its children are traversed recursively

Preorder TraversalAlgorithm

Algorithm preorder(Tv)Visit(v)for each child w of v dopreorder(Tw)

Preorder TraversalPreorder Print

public void preorderPrint(SimpleTree TPosition v)Systemoutprintln(Telement(v))Enumeration children_of_v=Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()preorderPrint(Tw)

Postorder TraversalDefinition

bull Recursively traverses the subtrees rooted at the children of the root first and then visits the root

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29

Basic Algorithms on Trees

Depth and Height

bull The height of an entire tree T is defined to be the height of the root of T Another way to view the height of a tree T is the maximum depth of a node of T which is achieved at one or more external nodes

Basic Algorithms on Trees

Depth and Height

public int height1(SimpleTree T)

int h

Enumberation nodes_of_T=Tpositions()

while(nodes_of_ThasMoreElements())

Position v=(Position)nodes_of_TnextElement()

if(TisExternal(v))

h=Mathmax(hdepth(Tv))

return h

The Trees Abstract Data Type

Example

Height Algorithm 1

Basic Algorithms on Trees

Depth and Height

public int height2(SimpleTree T Position v)if(TisExternal(v))return 0elseint h=0Enumeration children_of_v= Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()h=Mathmax(hTheight2(w))return 1+h

Preorder Traversal

Definition

bull A traversal of a tree T is a systematic way of accessing or ldquovisitingrdquo all the nodes of T

bull In a preorder traversal of a tree T the root of T is visited first and then the subtree rooted at its children are traversed recursively

Preorder TraversalAlgorithm

Algorithm preorder(Tv)Visit(v)for each child w of v dopreorder(Tw)

Preorder TraversalPreorder Print

public void preorderPrint(SimpleTree TPosition v)Systemoutprintln(Telement(v))Enumeration children_of_v=Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()preorderPrint(Tw)

Postorder TraversalDefinition

bull Recursively traverses the subtrees rooted at the children of the root first and then visits the root

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29

Basic Algorithms on Trees

Depth and Height

public int height1(SimpleTree T)

int h

Enumberation nodes_of_T=Tpositions()

while(nodes_of_ThasMoreElements())

Position v=(Position)nodes_of_TnextElement()

if(TisExternal(v))

h=Mathmax(hdepth(Tv))

return h

The Trees Abstract Data Type

Example

Height Algorithm 1

Basic Algorithms on Trees

Depth and Height

public int height2(SimpleTree T Position v)if(TisExternal(v))return 0elseint h=0Enumeration children_of_v= Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()h=Mathmax(hTheight2(w))return 1+h

Preorder Traversal

Definition

bull A traversal of a tree T is a systematic way of accessing or ldquovisitingrdquo all the nodes of T

bull In a preorder traversal of a tree T the root of T is visited first and then the subtree rooted at its children are traversed recursively

Preorder TraversalAlgorithm

Algorithm preorder(Tv)Visit(v)for each child w of v dopreorder(Tw)

Preorder TraversalPreorder Print

public void preorderPrint(SimpleTree TPosition v)Systemoutprintln(Telement(v))Enumeration children_of_v=Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()preorderPrint(Tw)

Postorder TraversalDefinition

bull Recursively traverses the subtrees rooted at the children of the root first and then visits the root

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29

The Trees Abstract Data Type

Example

Height Algorithm 1

Basic Algorithms on Trees

Depth and Height

public int height2(SimpleTree T Position v)if(TisExternal(v))return 0elseint h=0Enumeration children_of_v= Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()h=Mathmax(hTheight2(w))return 1+h

Preorder Traversal

Definition

bull A traversal of a tree T is a systematic way of accessing or ldquovisitingrdquo all the nodes of T

bull In a preorder traversal of a tree T the root of T is visited first and then the subtree rooted at its children are traversed recursively

Preorder TraversalAlgorithm

Algorithm preorder(Tv)Visit(v)for each child w of v dopreorder(Tw)

Preorder TraversalPreorder Print

public void preorderPrint(SimpleTree TPosition v)Systemoutprintln(Telement(v))Enumeration children_of_v=Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()preorderPrint(Tw)

Postorder TraversalDefinition

bull Recursively traverses the subtrees rooted at the children of the root first and then visits the root

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29

Basic Algorithms on Trees

Depth and Height

public int height2(SimpleTree T Position v)if(TisExternal(v))return 0elseint h=0Enumeration children_of_v= Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()h=Mathmax(hTheight2(w))return 1+h

Preorder Traversal

Definition

bull A traversal of a tree T is a systematic way of accessing or ldquovisitingrdquo all the nodes of T

bull In a preorder traversal of a tree T the root of T is visited first and then the subtree rooted at its children are traversed recursively

Preorder TraversalAlgorithm

Algorithm preorder(Tv)Visit(v)for each child w of v dopreorder(Tw)

Preorder TraversalPreorder Print

public void preorderPrint(SimpleTree TPosition v)Systemoutprintln(Telement(v))Enumeration children_of_v=Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()preorderPrint(Tw)

Postorder TraversalDefinition

bull Recursively traverses the subtrees rooted at the children of the root first and then visits the root

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29

Preorder Traversal

Definition

bull A traversal of a tree T is a systematic way of accessing or ldquovisitingrdquo all the nodes of T

bull In a preorder traversal of a tree T the root of T is visited first and then the subtree rooted at its children are traversed recursively

Preorder TraversalAlgorithm

Algorithm preorder(Tv)Visit(v)for each child w of v dopreorder(Tw)

Preorder TraversalPreorder Print

public void preorderPrint(SimpleTree TPosition v)Systemoutprintln(Telement(v))Enumeration children_of_v=Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()preorderPrint(Tw)

Postorder TraversalDefinition

bull Recursively traverses the subtrees rooted at the children of the root first and then visits the root

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29

Preorder TraversalAlgorithm

Algorithm preorder(Tv)Visit(v)for each child w of v dopreorder(Tw)

Preorder TraversalPreorder Print

public void preorderPrint(SimpleTree TPosition v)Systemoutprintln(Telement(v))Enumeration children_of_v=Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()preorderPrint(Tw)

Postorder TraversalDefinition

bull Recursively traverses the subtrees rooted at the children of the root first and then visits the root

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29

Preorder TraversalPreorder Print

public void preorderPrint(SimpleTree TPosition v)Systemoutprintln(Telement(v))Enumeration children_of_v=Tchildren(v)while(children_of_vhasMoreElements())Position w=(Position)children_of_vnextElement()preorderPrint(Tw)

Postorder TraversalDefinition

bull Recursively traverses the subtrees rooted at the children of the root first and then visits the root

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29

Postorder TraversalDefinition

bull Recursively traverses the subtrees rooted at the children of the root first and then visits the root

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29

Preorder TraversalAlgorithm

Algorithm postorder(Tv)for each child w of v dopreorder(Tw)Visit(v)

• Computer Science II
• Trees
• Slide 3
• Slide 4
• Slide 5
• The Trees Abstract Data Type
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Basic Algorithms on Trees
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Preorder Traversal
• Slide 26
• Slide 27
• Postorder Traversal
• Slide 29