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AVL Trees

B.Ramamurthy

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trees

static dynamic

game trees search trees priority queues and heaps

graphs

binary searchtrees

AVL trees2-3 trees tries Huffman

coding tree

Types of Trees

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Height of a Tree (Review)

Definition is same as level. Height of a tree is the length of the longest path from root to some leaf node. Height of a empty tree is -1.Height of a single node tree is 0.Recursive definition:

height(t) = -1 if T is empty = 0 if number of nodes = 1

= 1+ max(height(LT), height(RT)) otherwise

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Balanced Binary Search Trees

Let n be the number of nodes (keys) in a binary search tree and h be its height.Binary search tree offers a O(h) insert and delete if the tree is balanced. h is (log n) for a balanced tree. Height-balanced(k) tree has for every

node left subtree and right subtree differ in height at most by k.

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Unbalanced Tree

65

54

45

78

87

98

Maximum height

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Balanced Tree

87

78

98

87

87 87

Minimum Height

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AVL Property

AVL (Adelson, Velskii and Landis) tree is a height-balanced(1) tree that follows the property stated below: For every internal node v of the tree, the

heights of the children of v differ by at most 1.

AVL property maintains a logarithmic height in terms of the number of nodes of the tree thus achieving O(log n) for insert and delete.Lets look at some examples.

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AVL Tree: Example

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Non-AVL Tree

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Transforming into AVL Tree

Four different transformations are available called : rotationsRotations: single right, single left, double right, double leftThere is a close relationship between rotations and associative law of algebra.

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Transformations

Single right : ((T1 T2) T3) = (T1 (T2 T3)

Single left : (T1 (T2 T3)) = ((T1 T2) T3)

Double right : ((T1 (T2 T3)) T4) = ((T1 T2) (T3 T4))

Double left :(T1 ((T2 T3) T4)) = ((T1 T2) (T3 T4))B

C

C

A B A B

A

A

A

BB

B B

A

AA C

B C

A,B,C are nodes, Tn (n =1,2,3,4..) are subtrees

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AVL Balancing : Four Rotations

Single rightX3

X2

X1

X2

X1 X3

X1

X2

X3

Single left

X2

X3 X1 X3

X2

X1

Double rightX3

X2 X1

X3

X1 X2X2

X3

X1

Double left

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Example: AVL Tree for Airports

Consider inserting sequentially: ORY, JFK, BRU, DUS, ZRX, MEX, ORD, NRT, ARN, GLA, GCMBuild a binary-search treeBuild a AVL tree.

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Binary Search Tree for Airport Names

ORY

ZRHJFK

BRU MEX

ARNDUS

GLA

ORD

NRT

GCM

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An AVL Tree for Airport Names

ORY

JFK

BRU

Not AVL balanced

Single right JFK

BRU ORY

AVL Balanced

After insertion of ORY, JFK and BRU :

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An AVL Tree for Airport Names (contd.)After insertion of DUS, ZRH, MEX and ORD

JFK

BRU ORY

DUS MEX ZRH

ORD

Still AVL Balanced

After insertion of NRT?

JFK

BRU ORY

DUS MEX ZRH

ORD

NRT

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An AVL Tree …

JFK

BRU ORY

DUS MEX ZRH

ORD

NRT

Not AVL Balanaced

Double Left

JFK

BRU ORY

DUS NRT ZRH

ORDMEX

Now add ARN and GLA; noneed for rotations; Then add GCM

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An AVL Tree…

JFK

BRU

DUS

ORY

NRT ZRH

ORDMEX

ARN

GLA

GCM

NOT AVL BALANCED

JFK

BRU

GCM

ORY

NRT ZRH

ORDMEX

ARN

GLA

Double left

DUS

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Search Operation

For successful search, average number of comparisons:

sum of all (path length to node+1) / number of nodesFor the binary search tree (of airports) it is:

39/11 = 3.55For the AVL tree (of airports) it is :

33/11 = 3.0