AVL Trees - Data Structures - Lecture Slides, Slides of Data Structures and Algorithms

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Lecture 12

AVL Trees

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Types of Trees

trees

static dynamic

game trees search trees

priority queues

and heaps

graphs

binary search

trees

AVL trees

2-3 trees tries Huffman

coding

tree

AVL Trees

• Balanced binary search tree offer a O(log n) insert

and delete.

• But balancing itself costs O(n) in the average case.

• In this case, even though delete will be O(log n),

insert will be O(n).

• Is there any way to have a O(log n) insert too?

• Yes, by almost but not fully balancing the tree : AVL

(Adelson Velskii and Landis) balancing

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Height of a Tree

• 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) = 0 if number of nodes = 1

= -1 if T is empty

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

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AVL Tree: Example

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Non-AVL Tree

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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))

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Example: AVL Tree for Airports

• Consider inserting sequentially : ORY, JFK, BRU,

DUS, ZRX, MEX, ORD, NRT, ARN, GLA, GCM

• Build a binary-search tree

• Build a AVL tree.

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AVL Balancing : Four Rotations

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• X Single right
  • X - X - X - X1 X
• X - X - X - X Single left - X3 X - X - X - X - X Double right - X2 X - X - X1 X - X - X - X

An AVL Tree for Airport Names

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ORY

JFK

BRU

Not AVL balanced

Single right JFK

BRU

ORY

AVL Balanced

After insertion of ORY, JFK and BRU :

An AVL Tree …

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JFK

BRU

ORY

DUS MEX

ZRH

ORD

NRT

Not AVL Balanaced

Double Left

JFK

BRU

ORY

DUS NRT

ZRH

MEX ORD

Now add ARN and GLA; no

need for rotations; Then add GCM

An AVL Tree…

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JFK

BRU

DUS

ORY

NRT

ZRH

MEX ORD

ARN

GLA

GCM

NOT AVL BALANCED

JFK

BRU

GCM

ORY

NRT ZRH

MEX ORD

ARN

GLA

Double left

DUS

Known Performance Results of AVL trees

• AVL tree is a sub optimal solution.
• How to evaluate its performance?
• Bounds (upper bound and lower bound) for number of comparisons: C > log(n+1) + 1 C < 1.44 log(n+2) - 0.
• AVL trees are no more than 44% worse than optimal trees. 19

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