11.2 tree binary tree

7/23/2019 11.2 Tree Binary Tree

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CSG2A3ALGORITMA dan STRUKTUR DATA

Tree Data Structure



Binary Tree Data Structure

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Binary Tree

Tree Data Structure with maximum child (degree)of 2, which are referred to as the left child and theright child.

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Properties of Binary Tree

The number of nodes n a full binary tree is

– At least

–At most

– !here h is the height of the tree

"aximum #ode for each le$el % 2n

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Types of Binary Tree

'ull inary Tree

om*lete inary Tree

S+ewed inary Tree



Full Binary Tree

A tree in which e$ery node other than the lea$eshas two children

– sometimes called *ro*er binary tree or 2-tree



Complete Binary Tree

a binary tree in which e$ery le$el, exce*t *ossiblythe last, is com*letely filled, and all nodes are asfar left as *ossible

/

A A

A

A

D

A

D 0



Skewed Tree

inary Tree with an unbalanced branch betweenleft and right branch

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ADT Binary Tree

Array e*resentation

in+ed list re*resentation

4

id value

5 A

2

3

& D

0

'

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Array Representation

if a node has an index i, its children are found atindices

– eft child 2i + 1

– ight child 2i + 2

while its *arent (if any) is found at index

– (assuming the root has index 7ero)

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Array Representation

9roblem

– !aste s*ace

– :nsertion ; deletion *roblem

Array e*resentation is good forom*lete inary Tree ty*es

– inary <ea* Tree

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1 2 ! " # $ % & 1#

A - - - - D = 0



'inked 'ist Representation

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Ty*e infoty*e integerTy*e address *ointer to #ode

Ty*e #ode >info infoty*eleft addressright address

?

left :nfo right

Ty*e inTree address

Dictionaryroot inTree



Binary Tree ( Create )ew )odefunction create#ode( x infoty*e ) address

Algorithm

allocate( # )info( # ) x

left( # ) #ull

right( # ) #ull

#

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*+ample Application of Binary TreeArithmetic 0x*ression Tree

inary Search Tree

Decision Tree

A@ Tree

9riority ueue

– inary <ea* Tree

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Arit,metic *+pression TreeA s*ecific a**lication of a binary tree to e$aluatecertain ex*ressions

0xam*le – (a-b) ; ((cBd) C e)

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*+ercise - create t,e tree( a B b ) ; ( c d C e ) B f B g C h ; i

(( A B ) C ( B D )) ; ( 0 B ' C < )

( - (52 - (3 B /))) ;((5 B 8) B 2) C(2 C(3 B 5))

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Binary Searc, TreeErdered ; sorted binary tree

– :nternal nodes each store a +ey

– two distinguished sub-trees

the +ey in each node must be

– greater than all +eys stored in the left sub-tree

– and smaller than all +eys in right sub-tree

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Binary Searc, Tree ( .nsert new )ode9rocedure insertST( i x infoty*e, i;o # inTree )Algorithm

if ( # % #il ) then# create#ode( x )

elseif ( info( # ) ? x ) then

insertST( x , left( # ) )else if ( info( # ) > x ) then

insertST( x , right( # ) )else

out*ut(Fdu*licateG)

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Binary Searc, Tree ( Searc, )ode'unction find#ode( i x infoty*e, i;o # inTree ) addressAlgorithm

if ( info( # ) % x ) or ( # % #il ) then #

elseif ( info( # ) ? x ) then

find#ode( x , left( # ) )else if ( info( # ) > x ) then

find#ode( x , right( # ) )

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Binary Searc, Tree ( Delete )odeDelete node 9 - ogic

– 9 has no children remo$e 9

– 9 has 5 child re*lace 9 with its child

– 9 has 2 children

'ind where is either

– The leftmost child from the right sub-tree (successor of 9) or

– The rightmost child from the left sub-tree (*redecessor of 9)

e*lace 9 with

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Binary Searc, Tree ( Delete )ode'unction del"ostight( 9address, address ) address

Algorithmwhile ( right() >? #H ) do

right()info(9) info()left(9) deleteST( left(9), info() ) 9

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'unction del"osteft( 9address, address ) addressAlgorithm

while ( left() >? #H ) do left()info(9) info()right(9) deleteST( right(9), info() ) 9



Binary Searc, Tree ( Delete )ode'unction deleteST( 9 address, x infoty*e ) -? addressDictionary

'unction del"ostight( 9address, address ) address'unction del"osteft( 9address, address ) address

Algorithm

if( 9 % #H ) then 9

if (x > info(9)) thenleft(9) deleteST( left(9), x )

else if (x ? info(9)) thenright(9) deleteST( right(9), x )

elseif (left(9) >? #H ) then

9 del"ostight( 9 , left(9) )else if( right(9) >?#H ) then

9 del"osteft( 9, right(9) )else

delete 9 #H

9

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/uestion0



Traversal on Binary TreeD'S tra$ersal

– 9re-order

– :n-order

– 9ost-order

'S tra$ersal

– e$el-order

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Preorder TraversalDee* 'irst Search

oot eft ight

– 9refix notation

esult

– 'AD06:<

2



Preorder Traversal9rocedure *reErder( i;o root tree )

Algorithm

if ( root I% null ) then

out*ut( info( root ) )

*reErder( left( root ) )

*reErder( right( root ) )

2



.norder Traversaleft oot ight

– :nfix notation

esult – AD0'6<:

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.norder Traversal9rocedure inErder( i;o root tree )

Algorithm


inErder( left( root ) )


inErder( right( root ) )

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Postorder Traversaleft right oot

– 9ostfix notation

esult – A0D<:6'

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Postorder Traversal9rocedure *ostErder( i;o root tree )

Algorithm


*ostErder( left( root ) )

*ostErder( right( root ) )


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'evelorder Traversalreadth 'irst Search

recursi$ely at each node

esult

– '6AD:0<

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'evelorder Traversal9rocedure le$elErder( root tree )Dictionary

ueueAlgorithm

enJueue( , root )while ( not is0m*ty() )

n deJueue( )out*ut( n )enJueue( child ( n ) )

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*+ercise - write t,e traversal 1

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*+ercise - write t,e traversal 2

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*+ercise - write t,e traversal

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*+ercise - write t,e traversal !

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*+ercise - Create t,e TreeAssume there is E#0 tree, which if tra$ersed by:norder resulting 0AK'<D6, and whentra$ersed by 9reorder resulting 'A0KD<6

Draw the tree that satisfy the condition abo$e

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/uestion0



THANK YOU

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11.2 tree binary tree

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