07. AVL_Ballanced Binary Tree

8/8/2019 07. AVL_Ballanced Binary Tree

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(C) GOYANI MAHESH(C) GOYANI MAHESH 11

DATASTRUCTURES

MAHESH GOYANIM AHATMA G ANDHI I NSTITUE OF TECHNICAL EDUCATION & RESEARCH CENTER

[email protected]


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B ALANCEDB INARY

TREE

AVL TREE


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BST for 14, 15, 4, 9, 7, 18, 3, 5, 16, 20, 17

14

154

9

7

183

5

16 20

17

DEGENERATE B ST


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1415

4

97

18

3

5

16

20

17Linked List!

BST for 14, 15, 4, 9, 7, 18, 3, 5, 16, 20, 17

DEGENERATE B ST


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B ALANCED B ST

We should k eep th e tree balanc ed.On e idea would b e to hav e the left and right subtr ee s hav e th e sam e he ight

Does not forc e th e tree to b e shallow.

1415

18

16

20

174

97

3

5


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We could insist that eve ry nod e must hav e le ft and right subtr ee s of sam e he ight.

But this r eq uires that th e tree b e a com ple t e binary tr ee

To do this, th e re must hav e (2 d+1 1) data it ems, wh e re d is th e dep th of th e tree .

This is too rigid a condition.


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A Tr ee is called Balanc ed Binary Tr ee if eac h nod e poss esses on e of th e following prope rty:

A nod e is ca lled Le ft h ea vy if th e dep th of L eft sub tr ee is on e long e rthan Right sub tr ee

A node is ca lled Right h ea vy if th e dep th of Right sub tr ee is o ne longerthan Light sub tr ee

A node is ca lled b a lance d if th e dep th of both th e sub tr ee is s ame .

This tr ee is a lso k nown as AVL (Adelson Velskii and Landis) Tree

An AVL tree is iden tica l to a BST excep t

he ight of th e le ft an d right subtr ee s can diff e r by a t most 1.

he ight of an empty tr ee is d e fine d to b e ( 1).

Ba lance d Bina ry Tr ee is classified in to ca t egories :

He ight B a lance d

We ight B a lance d

AVL tree increa ses th e e fficienc y by r edu cing th e sea rching tim e in com pa re to bi na ry tr ee

TERMINOLOGY


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Graphical Representation

A

H

C

F

I

roo t

Right sub tr eeLeft sub tr ee

B

E

GG

D

B

R

L


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An AVL Tree An AVL Tree5

82

4

3

1 7

level0

1

2

3


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Not an AVL treeNot an AVL tree6

81

4

3

1

5

level0

1

2

3


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The heig ht of a b ina ry tr ee i s th e maxi mum l eve l of it s lea ves ( a lso ca lled t he dept h).

The b a lance of a node in a b ina ry t ree i s d e f ine d as t he heig ht of it s le f t sub t ree m inus h eig ht of it s r ig ht sub t ree .

He re , for exa mple , i s a b a lance d t ree . Eac h nod e has an indicate d b a lance of 1, 0, or1.

-1

01

0

0

0

0

-1

0

1

00 0

0 00 0


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If w e inse rt ne w node in AVL Tree , possibl e chan ges in st a tus of tr ee are as followi ng:

The nod e was e ithe r left or h ea vy an d h as now b ecom e b a lance d

The nod e was b a lance d an d h as now b ec om e left or right h ea vy

The nod e is h ea vy an d ne w node is inse rt ed in hea vy sub tr ee , this crea t esunb a lance d tr ee , su ch a node is ca lled critica l node

HEIGHT B ALANCED TREE


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HEIGHT B ALANCED TREE

A We ight b a lance d tr ee is a bina ry tr ee in which additio na l we ight fi e ld is a lso th ere .

The nod e of th e we ight b a lance d tr ee cont a ine d four fi e lds :

Da t a Elemen t

Le ft Child point er

Right Child Poi nt er

A Prob ability or We ight Fi e ld

A node , whi ch is most acce ssible is given max imum priority an d s e t as a root


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INSERT NODE

1. If this is first nod e th an a lloca t e node , s e t its fi e ld an dre tur n

2. If DATA is a lrea dy presen t in tree th an disca rd DATA, e lse link ne w node an d r e tur n

3. Sea rch u nb a lance d node

4. Adjust for u nb a lance f ac tor, if th e re is no critica l nod e ,re tur n

5. If node is b a lance d an d th an b ecom es h ea vy or th e nod e is h ea vy an d now b ecom es b a lance d th en adjust th e b a lance f ac tor

an d r e tur n

6. Reb a lance th e tree an d r e tur n.


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AVL TREE CREATION

Inse rt(1)Inse rt(1)1


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AVL TREE CREATION

Inse rt(2)Inse rt(2)1

2


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AVL TREE CREATION

Inse rt(3) si ngle le ft rot a tionInse rt(3) si ngle le ft rot a tion1

2

3

-2


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AVL TREE CREATION

Inse rt(3) si ngle le ft rot a tionInse rt(3) si ngle le ft rot a tion1

2

3

-2


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AVL TREE CREATION

Inse rt(3)Inse rt(3)

1

2

3


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AVL TREE CREATION

1

2

3

4


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AVL TREE CREATION

1

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5

-2



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AVL TREE CREATION


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AVL TREE CREATION


1

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AVL TREE CREATION


1

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AVL TREE CREATION


1

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7

-2


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AVL TREE CREATION


1

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5

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07. AVL_Ballanced Binary Tree

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