csc311: data structures 1 chapter 7: trees objectives: introduce general tree structure and tree adt...

CSC311: Data StructuresCSC311: Data Structures 11

Chapter 7: TreesChapter 7: TreesObjectives:Objectives:

Introduce general tree structure and Introduce general tree structure and Tree ADTTree ADT

Discuss the depth and height of treesDiscuss the depth and height of trees Introduce the tree traversal algorithms Introduce the tree traversal algorithms Specialize to binary treesSpecialize to binary trees Implement binary trees with linked Implement binary trees with linked

structure and array-list structurestructure and array-list structure Introduce the Template Method PatternIntroduce the Template Method Pattern

TreesTrees CSC311: Data StructuresCSC311: Data Structures 22

What is a TreeWhat is a TreeIn computer science, a In computer science, a tree is an abstract tree is an abstract model of a hierarchical model of a hierarchical structurestructure

A tree consists of nodes A tree consists of nodes with a parent-child with a parent-child relationrelation

Applications:Applications:– Organization chartsOrganization charts– File systemsFile systems– Programming Programming

environmentsenvironments

Computers”R”Us

Sales R&DManufacturing

Laptops DesktopsUS International

Europe Asia Canada

TreesTrees CSC311: Data StructuresCSC311: Data Structures 33subtree

Tree TerminologyTree TerminologyRoot: node without parent (A)Root: node without parent (A)Internal node: node with at Internal node: node with at least one child (A, B, C, F)least one child (A, B, C, F)External node (a.k.a. leaf ): External node (a.k.a. leaf ): node without children (E, I, J, K, node without children (E, I, J, K, G, H, D)G, H, D)Ancestors of a node: parent, Ancestors of a node: parent, grandparent, grand-grandparent, grand-grandparent, etc.grandparent, etc.Depth of a node: number of Depth of a node: number of ancestorsancestorsHeight of a tree: maximum Height of a tree: maximum depth of any node (3)depth of any node (3)Descendant of a node: child, Descendant of a node: child, grandchild, grand-grandchild, grandchild, grand-grandchild, etc.etc.

A

B DC

G HE F

I J K

Subtree: tree Subtree: tree consisting of a node consisting of a node and its descendantsand its descendants


Tree ADT Tree ADT We use positions to We use positions to abstract nodesabstract nodes

Generic methods:Generic methods:– integer integer sizesize()()– boolean boolean isEmptyisEmpty()()– Iterator Iterator iteratoriterator()()– Iterator Iterator positionspositions()()

Accessor methods:Accessor methods:– position position rootroot()()– position position parentparent(p)(p)– positionIterator positionIterator

childrenchildren(p)(p)

Query methods:Query methods:– boolean boolean isInternalisInternal(p)(p)– boolean boolean isExternalisExternal(p)(p)– boolean boolean isRootisRoot(p)(p)

Update method:Update method:– object object replace replace (p, o)(p, o)

Additional update Additional update methods may be defined methods may be defined by data structures by data structures implementing the Tree implementing the Tree ADTADT


Depth of a NodeDepth of a Node

The depth of a The depth of a node v in a tree T node v in a tree T is defined as:is defined as:– If v is the root, then If v is the root, then

its depth is 0;its depth is 0;– Otherwise, its Otherwise, its

depth is one plus depth is one plus the depth of its the depth of its parent parent

Algorithm depth(T, v)Algorithm depth(T, v)Input: Tree T and a node vInput: Tree T and a node v

Output: an integer that is the Output: an integer that is the depth of v in Tdepth of v in T

If v is the root of T thenIf v is the root of T then

return 0return 0

ElseElse

return 1 + depth(T, parent(v))return 1 + depth(T, parent(v))


Height of a NodeHeight of a Node

The height of a The height of a node v in a tree T node v in a tree T is defined as:is defined as:– If v is an external, If v is an external,

then its height is 0;then its height is 0;– Otherwise, its Otherwise, its

height is one plus height is one plus the maximum the maximum height of its height of its children children

Algorithm height(T, v)Algorithm height(T, v)Input: Tree T and a node vInput: Tree T and a node v

Output: an integer that is the Output: an integer that is the height of v in Theight of v in T

If v is an external of T thenIf v is an external of T then

return 0return 0

ElseElse

hh00

for each child w of v in T dofor each child w of v in T do

hhmax(h, height(T,w))max(h, height(T,w))

return 1+hreturn 1+h


Features on HeightFeatures on Height

1.1. The height of a nonempty tree T is The height of a nonempty tree T is equal to the maximum depth of an equal to the maximum depth of an external node of Texternal node of T

2.2. Let T be a tree with n nodes, and let Let T be a tree with n nodes, and let ccvv denote the number if children of denote the number if children of a node v of T. The summing over a node v of T. The summing over the vertices in T, the vertices in T, vvccvv=n-1.=n-1.


Preorder TraversalPreorder TraversalA traversal visits the nodes of A traversal visits the nodes of a tree in a systematic mannera tree in a systematic manner

In a preorder traversal, a In a preorder traversal, a node is visited before its node is visited before its descendants descendants

Application: print a structured Application: print a structured documentdocument

Make Money Fast!

1. Motivations References2. Methods

2.1 StockFraud

2.2 PonziScheme

1.1 Greed 1.2 Avidity2.3 BankRobbery

1

2

3

5

4 6 7 8

9

Algorithm preOrder(v)visit(v)for each child w of v

preorder (w)


Postorder TraversalPostorder TraversalIn a postorder traversal, a In a postorder traversal, a node is visited after its node is visited after its descendantsdescendants

Application: compute space Application: compute space used by files in a directory used by files in a directory and its subdirectoriesand its subdirectories

Algorithm postOrder(v)for each child w of v

postOrder (w)visit(v)

cs16/

homeworks/todo.txt

1Kprograms/

DDR.java10K

Stocks.java25K

h1c.doc3K

h1nc.doc2K

Robot.java20K

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3

1

7

2 4 5 6

8


Binary TreesBinary TreesA binary tree is a tree with the A binary tree is a tree with the following properties:following properties:– Each internal node has at most Each internal node has at most

two children (exactly two for two children (exactly two for properproper binary trees) binary trees)

– The children of a node are an The children of a node are an ordered pairordered pair

We call the children of an We call the children of an internal node left child and right internal node left child and right childchildAlternative recursive definition: Alternative recursive definition: a binary tree is eithera binary tree is either– a tree consisting of a single a tree consisting of a single

node, ornode, or– a tree whose root has an a tree whose root has an

ordered pair of children, each of ordered pair of children, each of which is a binary treewhich is a binary tree

Applications:Applications:– arithmetic arithmetic

expressionsexpressions– decision processesdecision processes– searchingsearching

A

B C

F GD E

H I


Arithmetic Expression TreeArithmetic Expression TreeBinary tree associated with an arithmetic Binary tree associated with an arithmetic expressionexpression– internal nodes: operatorsinternal nodes: operators– external nodes: operandsexternal nodes: operands

Example: arithmetic expression tree for the Example: arithmetic expression tree for the expression (2 expression (2 ((a a 1) 1) (3 (3 b))b))

2

a 1

3 b


Decision TreeDecision Tree

Binary tree associated with a decision processBinary tree associated with a decision process– internal nodes: questions with yes/no answerinternal nodes: questions with yes/no answer– external nodes: decisionsexternal nodes: decisions

Example: dining decisionExample: dining decision

Want a fast meal?

How about coffee? On expense account?

Starbucks Spike’s Al Forno Café Paragon

Yes No

Yes No Yes No


BinaryTree ADTBinaryTree ADTThe BinaryTree ADT The BinaryTree ADT extends the Tree extends the Tree ADT, i.e., it inherits ADT, i.e., it inherits all the methods of all the methods of the Tree ADTthe Tree ADT

Additional methods:Additional methods:– position position leftleft(p)(p)– position position rightright(p)(p)– boolean boolean hasLefthasLeft(p)(p)– boolean boolean hasRighthasRight(p)(p)

Update methods Update methods may be defined may be defined by data structures by data structures implementing the implementing the BinaryTree ADTBinaryTree ADT


Properties of Proper Binary TreesProperties of Proper Binary Trees

NotationNotationnn number of number of

nodesnodes

ee number of number of external nodesexternal nodes

ii number of number of internal nodesinternal nodes

hh heightheight

Properties:Properties:– e e i i 11

– n n 22e e 11– h h ii

– h h ((n n 1)1)22– e e 22hh

– h h loglog22 ee

– h h loglog22 ( (n n 1)1) 11


Inorder TraversalInorder TraversalIn an inorder traversal a In an inorder traversal a node is visited after its node is visited after its left subtree and before its left subtree and before its right subtreeright subtreeApplication: draw a binary Application: draw a binary treetree– x(v) = inorder rank of vx(v) = inorder rank of v– y(v) = depth of vy(v) = depth of v

Algorithm inOrder(v)if hasLeft (v)

inOrder (left (v))visit(v)if hasRight (v)

inOrder (right (v))

3

1

2

5

6

7 9

8

4


Print Arithmetic ExpressionsPrint Arithmetic ExpressionsSpecialization of an Specialization of an inorder traversalinorder traversal– print operand or operator print operand or operator

when visiting nodewhen visiting node– print “(“ before traversing print “(“ before traversing

left subtreeleft subtree– print “)“ after traversing print “)“ after traversing

right subtreeright subtree

Algorithm printExpression(v)if hasLeft (v)

print(“(’’)inOrder (left(v))

print(v.element ())if hasRight (v)

inOrder (right(v))print (“)’’)

2

a 1

3 b((2 (a 1)) (3 b))


Evaluate Arithmetic Evaluate Arithmetic ExpressionsExpressions

Specialization of a Specialization of a postorder traversalpostorder traversal– recursive method recursive method

returning the value of a returning the value of a subtreesubtree

– when visiting an internal when visiting an internal node, combine the values node, combine the values of the subtreesof the subtrees

Algorithm evalExpr(v)if isExternal (v)

return v.element ()else

x evalExpr(leftChild (v))

y evalExpr(rightChild (v))

operator stored at vreturn x y

2

5 1

3 2


Euler Tour TraversalEuler Tour TraversalGeneric traversal of a binary treeGeneric traversal of a binary tree

Includes a special cases the preorder, postorder and inorder Includes a special cases the preorder, postorder and inorder traversalstraversals

Walk around the tree and visit each node three times:Walk around the tree and visit each node three times:– on the left (preorder)on the left (preorder)– from below (inorder)from below (inorder)– on the right (postorder)on the right (postorder)

2

5 1

3 2

LB

R


Template Method PatternTemplate Method PatternGeneric algorithm that Generic algorithm that can be specialized by can be specialized by redefining certain stepsredefining certain stepsImplemented by means Implemented by means of an abstract Java class of an abstract Java class Visit methods that can be Visit methods that can be redefined by subclassesredefined by subclassesTemplate method Template method eulerToureulerTour– Recursively called on the Recursively called on the

left and right childrenleft and right children– A A ResultResult object with fields object with fields

leftResultleftResult,, rightResult rightResult andand finalResultfinalResult keeps track of keeps track of the output of the the output of the recursive calls to recursive calls to eulerToureulerTour

public abstract class EulerTour {protected BinaryTree tree;protected void visitExternal(Position p, Result r) { }protected void visitLeft(Position p, Result r) { }protected void visitBelow(Position p, Result r) { }

protected void visitRight(Position p, Result r) { } protected Object eulerTour(Position p) {

Result r = new Result();if tree.isExternal(p) { visitExternal(p, r); }

else {visitLeft(p, r);r.leftResult = eulerTour(tree.left(p));visitBelow(p, r);r.rightResult = eulerTour(tree.right(p));visitRight(p, r);return r.finalResult;

} …


Specializations of EulerTourSpecializations of EulerTour

We show how to We show how to specialize class specialize class EulerTour to EulerTour to evaluate an evaluate an arithmetic arithmetic expressionexpression

AssumptionsAssumptions– External nodes External nodes

store Integer store Integer objectsobjects

– Internal nodes Internal nodes store store OperatorOperator objects objects supporting methodsupporting method

operation operation (Integer, Integer) (Integer, Integer)

public class EvaluateExpressionextends EulerTour {

protected void visitExternal(Position p, Result r) {r.finalResult = (Integer) p.element();

}

protected void visitRight(Position p, Result r) {Operator op = (Operator) p.element();r.finalResult = op.operation(

(Integer) r.leftResult,(Integer) r.rightResult);

}

…

}


Linked Structure for TreesLinked Structure for TreesA node is represented by A node is represented by an object storingan object storing– ElementElement– Parent nodeParent node– Sequence of children Sequence of children

nodesnodes

Node objects implement Node objects implement the Position ADTthe Position ADT

B

DA

C E

F

B

A D F

C

E


Linked Structure for Binary TreesLinked Structure for Binary TreesA node is represented A node is represented by an object storingby an object storing– ElementElement– Parent nodeParent node– Left child nodeLeft child node– Right child nodeRight child node

Node objects Node objects implement the Position implement the Position ADTADT

B

DA

C E

B

A D

C E


Array-Based Representation Array-Based Representation of Binary Treesof Binary Trees

nodes are stored in an arraynodes are stored in an array

…

let rank(node) be defined as follows: rank(root) = 1 if node is the left child of parent(node),

rank(node) = 2*rank(parent(node)) if node is the right child of parent(node),

rank(node) = 2*rank(parent(node))+1

1

2 3

6 74 5

10 11

A

HG

FE

D

C

B

J


Array-Based Representation of Array-Based Representation of Binary Trees: PropertiesBinary Trees: Properties

Let n be the number of nodes of a binary tree T Let pM be the maximum value of rank(v) over all the

nodes of T The array size N = pM+1

In the worst case, N = 2n

csc311: data structures 1 chapter 7: trees objectives: introduce general tree structure and tree adt...

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