DATA STRUCTURE

TREEDisusun Oleh : Budi Arifitama

Pertemuan ke-8

OBJECTIVES• Define trees as data structures• Define the terms associated with trees• Discuss tree traversal algorithms• Discuss Binary Tree• Examine a binary tree example

SUB TOPIC Tree Introduction Term Associated With Tree Binary Tree Traverse Tree

TREES• A tree is a nonlinear data structure

used to represent entities that are in some hierarchical relationship

• Examples in real life: • Family tree• Table of contents of a book• Class inheritance hierarchy in Java• Computer file system (folders and subfolders)• Decision trees• Top-down design

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EXAMPLE: COMPUTER FILE SYSTEM

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Root directory of C drive

Documents and Settings Program Files My Music

Desktop Favorites Start Menu Microsoft OfficeAdobe

TREE EXAMPLE

TREE DEFINITION• Tree: a set of elements of the same type such

that• It is empty• Or, it has a distinguished element

called the root from which descend zero or more trees (subtrees)

• What kind of definition is this?• What is the base case?• What is the recursive part?

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TREE TERMINOLOGY

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Leaf nodes

RootInterior nodes

TREE TERMINOLOGY• Nodes: the elements in the tree• Edges: connections between nodes• Root: the distinguished element that is the

origin of the tree• There is only one root node in a tree

• Leaf node: a node without an edge to another node

• Interior node: a node that is not a leaf node• Empty tree has no nodes and no edges

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• Parent or predecessor: the node directly above in the hierarchy• A node can have only one parent

• Child or successor: a node directly below in the hierarchy

• Siblings: nodes that have the same parent• Ancestors of a node: its parent, the

parent of its parent, etc.• Descendants of a node: its children, the

children of its children, etc.

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Tree Terminology

DISCUSSION• Does a leaf node have any children?• Does the root node have a parent?• How many parents does every node other

than the root node have?

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HEIGHT OF A TREE• A path is a sequence of edges leading

from one node to another• Length of a path: number of edges on

the path• Height of a (non-empty) tree : length

of the longest path from the root to a leaf• What is the height of a tree that has only a

root node?• By convention, the height of an empty

tree is -1

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LEVEL OF A NODE• Level of a node : number of edges

between root and node• It can be defined recursively:

• Level of root node is 0• Level of a node that is not the root node is

level of its parent + 1• Question: What is the level of a node in

terms of path length?• Question: What is the height of a tree in

terms of levels?

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LEVEL OF A NODE

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Level 0

Level 1

Level 2

Level 3

SUBTREES• Subtree of a node: consists of a child

node and all its descendants

• A subtree is itself a tree• A node may have many

subtrees

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SUBTREES

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Subtrees of the root node

SUBTREES

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Subtrees of the node labeled E

E

MORE TREE TERMINOLOGY• Degree or arity of a node: the number of

children it has

• Degree or arity of a tree: the maximum of the degrees of the tree’s nodes

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TREE TERMINOLGY

LATIHANAncestor (F)?Predesesor (F) ?Descendant (B)?Succesor (B)?Parent (I)?Child (C)?Sibling (G)?Size?Height?Root?Leaf?Degree (C)?

LATIHANGambarkan tree dari representasi berikut :

DFDBKJKLBABCH DHKFEFGJI

Tentukan :1. Root2. Leaf3. Heigth4. Child H5. Parent A

BINARY TREE

Finite (possibly empty) collection of elements. A nonempty binary tree has a root element. The remaining elements (if any) are partitioned

into two binary trees. These are called the left and right subtrees of

the binary tree.

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BINARY TREE There are many variations on trees but

we will start with binary trees binary tree: each node has at most two

children the possible children are usually referred to

as the left child and the right child

parent

left child right child

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FULL BINARY TREE full binary tree: a binary tree is which

each node was exactly 2 or 0 children

QUESTION What is the maximum height of a full

binary tree with 11 nodes?A. 1B. 3C. 5D. 7E. Not possible to construct a full binary

tree with 11 nodes.

CS314Binary Trees 25

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COMPLETE BINARY TREE complete binary tree: a binary tree in

which every level, except possibly the deepest is completely filled. At depth n, the height of the tree, all nodes are as far left as possible

Where would the next node goto maintain a complete tree?

Binary Trees CS314

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PERFECT BINARY TREE perfect binary tree: a binary tree with all

leaf nodes at the same depth. All internal nodes have exactly two children.

a perfect binary tree has the maximum number of nodes for a given height

a perfect binary tree has (2(n+1) - 1) nodes where n is the height of the tree height = 0 -> 1 node height = 1 -> 3 nodes height = 2 -> 7 nodes height = 3 -> 15 nodes

Binary Trees CS314

DIFFERENCES BETWEEN A TREE & A BINARY TREE

No node in a binary tree may have a degree more than 2, whereas there is no limit on the degree of a node in a tree.

A binary tree may be empty; a tree cannot be empty.

DIFFERENCES BETWEEN A TREE & A BINARY TREE

The subtrees of a binary tree are ordered; those of a tree are not ordered.

a

b

a

b

• Are different when viewed as binary trees.• Are the same when viewed as trees.

CONTOH BINARY TREE Representasi ekspresi arithmatik

MINIMUM NUMBER OF NODES Minimum number of nodes in a binary tree whose

height is h. At least one node at each of first h levels.

minimum number of nodes is h

MAXIMUM NUMBER OF NODES

All possible nodes at first h levels are present.

Maximum number of nodes

= 1 + 2 + 4 + 8 + … + 2h-1

= 2h - 1

REPRESENTATION OF BINARY TREE A binary tree can bee represented by an

Array or a linked list.

REPRESENTASI TREE (ARRAY)

H D K B F J L A C E G I

Array

0 1 2 3 4 5 6 7 8 9 10 11

LINKED REPRESENTATIONH

KD

B

A

LJ

I

leftChild element rightChild

root

F

CE G

ARRAY REPRESENTATION

tree[] 5 10

a b c d e f g h i j

b

a

c

d e f g

h i j

1

2 3

4 5 6 7

8 9 10

0

LATIHAN

a

b

1

3

c7

d15

RIGHT-SKEWED BINARY TREE

a

b

1

3

c7

d15

tree[] 0 5 10

a - b - - - c - - - - - - -15d

BINARY TREE TRAVERSAL

DEFINISI Penelusuran seluruh node pada binary

tree. Metode :

Preorder InorderPostorderLevel order

PREORDER TRAVERSAL Preorder traversal

1. Cetak data pada root2. Secara rekursif mencetak seluruh data

pada subpohon kiri3. Secara rekursif mencetak seluruh data

pada subpohon kanan

PREORDER EXAMPLE (VISIT = PRINT)

a

b c

a b c

PREORDER EXAMPLE (VISIT = PRINT)

a

b c

d ef

g h i j

a b d g h e i c f j

PREORDER OF EXPRESSION TREE

+

a b

-

c d

+

e f

*

/

Gives prefix form of expression!

/ * + a b - c d + e f

INORDER TRAVERSAL Inorder traversal

1.Secara rekursif mencetak seluruh data pada subpohon kiri

2.Cetak data pada root3.Secara rekursif mencetak seluruh data pada

subpohon kanan

INORDER EXAMPLE (VISIT = PRINT)

a

b c

b a c

INORDER EXAMPLE (VISIT = PRINT)

a

b c

d ef

g h i j

g d h b e i a f j c

POSTORDER TRAVERSAL Postorder traversal

1.Secara rekursif mencetak seluruh data pada subpohon kiri

2.Secara rekursif mencetak seluruh data pada subpohon kanan

3.Cetak data pada root

POSTORDER EXAMPLE (VISIT = PRINT)

a

b c

b c a

POSTORDER EXAMPLE (VISIT = PRINT)

a

b c

d ef

g h i j

g h d i e b j f c a

POSTORDER OF EXPRESSION TREE

+

a b

-

c d

+

e f

*

/

Gives postfix form of expression!

a b + c d - * e f + /

LATIHAN Telusuri pohon biner berikut dengan

menggunakan metode pre, in, post, dan level traversal.

LATIHAN 1

a. b.

+

*

3 5

-

2 /

8 4

LATIHAN 2