TREE (JAVA)

8/3/2019 TREE (JAVA)

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CHAPTER 6 :CHAPTER 6 : TREETREE

6.1 INTRODUCTION

6.1.1 What is Tree?6.1.2 What is Binary Tree?. .

6.1.4 Tree Traversal

.6.2.1 What is Expression Tree6.2.2 Building the expression tree

6.2.3 Traversing the expression tree6.2.4 Evaluating the expression value

1Intersession May 2009 UiTMT (MP, EZ, NIK)


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Introduction to TreeIntroduction to TreeWhat is Tree ?

A tree is a non-linearor two-dimensional data structurewith special properties.

Tree nodes contains two or more link

(Binary Tree :contains nodes with two links)

A tree can be divided into three subset :

root

Second subset left sub tree

Third subset right sub tree left subtree right subtree



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IntroductionIntroduction to Tree (cont)to Tree (cont)

Tree com onents re:

Root node first node in a tree.Child each link in the root node refers to.

tree)

Right child- first node in right sub tree (root node of rightsub tree)

Siblings nodes in the branch and same level.

Lea node a node with no children.



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Binary TreeBinary Tree

What is Binary TreeBT can have maximum of two child (branches) only .

Left sub tree is built on the left side of a node

Binary tree graphical representation

Right sub tree is built on the right side of a node

oo no e

BRight SubtreeLeft Subtree

CNode A and D are siblings

Node A and C is a Leaf


Node C is Child node of D


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Binary Tree (cont)Binary Tree (cont)

Example : ABC Binary Tree

Descriptions :

B C

-

- node B is the left subtree of node A

- node C is the right subtree of node A-

eve

Level 1

G

F

H I

(eg : no right subtree for node E)

- node D, G, H, I is a leaf (has null left

and ri ht subtree)

Level 2

- the depth of Tree ABC is 3 (max level,where first level starts with 0)

Level 3ABC



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Binary Tree TypesBinary Tree Types

a) Strictly Binary Tree (SBT)

Each node thats not a leaf has both left and right subtree

Formula to calculate the number of nodes :2n-1 where n is number of levels

Example : 2 (4) 1 7 nodes

A Level 0

B C

D E

Level 1

GF

Level 3



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Binary Tree TypesBinary Tree Types ((cont)cont)

b) Full Binary Tree (FBT)

A tree with leaves onl at the last level.

Example ofleaf: Node H, I, J, K, L, M, N, OLevel 0

Level 1C

Level 2D E F G

Level 3

IH ONMLKJ



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Binary Tree TypesBinary Tree Types ((cont)cont)

Formula to calculate the number of nodes in FBT:

,

have 2 x M nodes.Example : if level 2 has 4 nodes then

eve as x no es

b) At Level L, each FBT has 2L nodesExample :

at level 3 there will be 23 8 nodesc) Total nodes in FBT is addition of all nodes at each level.

Example : with 3 levelere are + + + no es



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Tree TraversalTree Traversal

A tree data structure can store many information in a particular

order to access and store.

Accessing data in tree data structure is called tree traversal.

Tree traversal is done recursively. Whenever a subtree is met,

the traversal order starts again

There are three ways to traverse the tree

Inordertraversal (left, root, right)

reor er raversa roo , e , r gPostordertraversal (left, right, root)



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Tree Traversal (cont)Tree Traversal (cont)

INORDER traversal

For each tree or subtree traversal starts

1. fromLEFTnode,

3. theRIGHTnode

A

B C

D E

Example : D B E A C



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PREORDER traversal


1. fromROOTnode,

3. theRIGHTnode

A

B C

Example : A B D E C



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POSTORDER traversal


1. fromLEFTnode,

3. theROOTnode

B C

D E

Example : D E B C A



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Example (a) :

B C

D F GE

H

nor er Preorder A B D H E C F G

Postorder H D E B F G C A



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Example (b) :

Q R

W

X

Y

nor er Preorder P Q W Y X R

Postorder Y W X Q R P



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ExpressionExpression TreeTree

What is Expression Tree

one of t e app ications of Binary Tree (BT)

used to re resent arithmetic ex ression

(infix, prefix and postfix)

.

expression must be converted to prefix or postfix

expression in order to build the expression tree

prefix or postfix expression will produce the same expression

tree if it comesfrom the same infix expression



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Building ExpressionBuilding Expression TreeTree

Building Expression Tree

a) Using prefix expression

Readprefix expression from LEFT to RIGHT.

Any operator must be at the ROOT, and any operand

must be at the leaf

Always open the LEFT branch first until no more branch

can be opened. Then open the RIGHT branch.



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Building ExpressionBuilding Expression Tree (Tree (cont)cont)

Example (a) :Infix => (A + B) * C * ( D - E)

prefix => * * + A B C D E

Le t Ri ht

*

* -

+ D E

A B

C



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Example (b) :refix => + A * - B C $ D * E F

Left Right

+

A *

- $

*

B

E F



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Building ExpressionBuilding Expression Tree (cont)Tree (cont)

b Usin ost ix ex ression

Readpostfix expression from RIGHT to LEFT.

be at the leaf

Always open the RIGHT branch first until no more branch. .



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Example (a) :Infix => A + B * C * D - E

postfix => A B + C * D E - *

* -

+ D E

A

C



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Example (b) :

postfix => A B C D E F * $ * +Left Right

+

A *

- $

DB C*



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ExpressionExpression TreeTree TraversalTraversal

Traversing The Expression Tree

Applying a different traversing technique to an expressiontree will result a different mathematical expression.

There are three traversing technique :

INORDER Traversal gives infix expression

(LEFT, ROOT, RIGHT) (OPERAND,OPERATOR,OPERAND)PREORDER Traversal gives prefix expression

, , , ,POSTORDER Traversal gives postfix expression

(LEFT, RIGHT, ROOT) (OPERAND,OPERAND,OPERATOR)



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Expression TreeExpression Tree Traversal (cont)Traversal (cont)

Example (a) :

*

* -

+ D EC

B

=> + * * -

Preorder => * * + A B C D EPostorder => A B + C * D E - *



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Expression TreeExpression Tree Traversal (cont)Traversal (cont)

Example (b) :

-F

C

$ +

+

BA

Inorder => A + B $ C D + E $ FPreorder => $ - $ + A B C + D E FPostorder => A B + C D E + - F



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Expression TreeExpression Tree EvaluationEvaluation

Evaluating Expression Tree

Steps :

Replace leaf(operand or variable) with the given value

Start with LEFT leaf

then OPERATOR andor every su tree

Evaluate the expression tree using the arithmetic process.



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Expression TreeExpression Tree Evaluation (cont)Evaluation (cont)

Example (a) :

144

$

-F

212

$ $16 4

DC

B

+

A

C

2 2

2 2 2



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Expression TreeExpression Tree Evaluation (cont)Evaluation (cont)

Example (b) :

if A= 5, B = 4, C=3, D=2, E=1 then RESULT is -19

-19

-

A*

524

C

*B

$ 2

46

ED 2 1


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