DS Lec 13 AVL

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AVL Trees

Lecture # 13

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Problem with the BST

Insert the nodes 1,2,3,4,5,6,7 in BST

Then search an element

How man o!erations the search will !er"orm here are the ad$anta%es o" BST&

7/23/2019 DS Lec 13 AVL


Balanced Trees

'rdered (inar trees are "airl eas to im!lement)

But, the can ha$e one draw(ac*

+$en an ordered (inar tree can de%rade into a linear

list i" the insertion routine recei$es elements inascendin% or nearl ascendin%- order)

1, 2, 3, 4 1

2

3

4

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Balanced trees

Se$eral methods ha$e (een de$elo!ed to

!re$ent trees "rom (ecomin% s*ewed)

Some o" the most !ower"ul is called ./L

trees)

Their name is deri$ed "rom the scientists who

"irst studied them0 .delson/els*ii and

Landis 162-

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AVL Trees

. (alanced (inar search tree 'ldest "orm o" (alanced (inar tree

.$oids s*ewin% ( re(alancin% when needed

Sel"(alancin%

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AVL Tree (Balance Binary Search

Tree) It is a (inar search tree in which the height of the

two sub-trees of every node never differ by

more than 1)

What about these trees?

3

6

82

1 4 7

3

6

82

1 4

5

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Building an AVL Tree ew node can (e inserted into the ri%ht or le"t su(tree as a!!ro!riate)

ew node can (e inserted without chan%in% the hei%ht o" the su(tree, in

which case neither the hei%ht nor the (alance o" the root will (e

chan%ed)

The onl case that can cause di""icult occurs when the new node is

added to a su(tree o" the root that is strictl taller than the othersu(tree, and the hei%ht is increased) This would cause one su(tree to

ha$e hei%ht 2 more than the other destroin% the ./L tree)

k 1

k 2

T1 T2T3

hh h+1

Total Height = h+3

k 2

k 1

T3T1 T2h

h+1

Total Height = h+2

Rotate Left

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Building an AVL Tree

BalanceHei%ht in"ormation can (e *e!t at the node)

hen we do insertion, we need to u!date all the

(alancin% in"ormation "or the nodes on the !ath

(ac* to the root)

k 1

k 2X

Y Z

k 2

k 1

X Y

Z

Equal o !ot"

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Building an AVL TreeSingle Rotation

E#a$%le& i'(et 1) 2) 3) 4) 5) 6) 7

1

2

3

Rotate2

1 3

*'(et 4) 5

2

1 3

4

5

Rotate

2

1 4

3 5

6

*'(et 6

Rotate

3

4

2 5

61

7*'(et 7

Rotate

3

4

2

5

61

7

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Insertion/ Rotation

E#a$%le& *'(et 3 i' the folloi'g tee a', $ake it -.L

4

6

2 8

1

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Building an AVL TreeInsertion/ Rotation

E#a$%le& *'(et ke/( 8 though 15 i' e0e(e o,e i' the

folloi'g tee a', $ake it a' -.L

3

4

2

5

61

7*'(et 15

!o ole$15

14*'(et 14

Rotate 3

4

2

5

61

15

14

7

The Rotatio' ha( 'ot fi#e, the height i$ala'e

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Double Right-Left Rotation

k 3

-

k 1

k 2

1

2Right Left

oule Rotatio'

k 2

-

k 1

k 3

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Double Right-Left Rotation

Right Left

oule Rotatio'

2

3

4

2

5

61

7

15

14

k 3

k 1

k 2

1

3

4

2

5

61

14

157

k 2

k 1k 3

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Building an AVL TreeDouble Left-Right Rotation

k 3

-

k 2

k 1

1

2 Left Right

oule Rotatio'

k 2

-

k 3

k 1

Left Right

oule Rotatio'1

2

23

25

29

2615

k 2

k 1

k 3

22

22

23

29

2515

k 2

k 1k 3

26

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ow we write a "ormal al%orithm to decide

which t!e o" rotation we !er"orm when ./L

condition is $iolated)

Let is a !ointer to the node at which ./Lcondition is $iolated)

ildi

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Building an AVL TreeIf P is left higher (Its left height is greater)

{ Q = P -> left;

if(Q is Right higher (Its right height is greater) //Perform Left-Right double rotation

{ Perform left rotation at QPerform right rotation at P

}

else erform right rotation at P //Perform single right rotation

}

else //P is right higher

{ Q = P -> right

if(Q is left higher (its left height is greater) //Perform Right-Left double rotation{ Perform right rotation at Q

Perform left rotation at P

}

else //erform single left rotation

erform left rotation at P

}

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Exercise

Insert these *es and ma*e an ./L tree 1,2,3,4,5,6,7,15,14,13,12,11,1,,

DS Lec 13 AVL

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