AVL Trees

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AVL Trees

• B.Ramamurthy

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Types of Trees
trees
dynamic
static
game trees
search trees
priority queues and heaps
graphs
Huffman coding tree
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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
45
54
65
Maximum height
78
87
98
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Balanced Tree
78
87
87
98
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 rotations
• Rotations single right, single left, double
  right, double left
• There 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))

A
A
B
B
A
A
B
B
C
B
A
A
C
B
B
C
A
A
B
C
A,B,C are nodes, Tn (n 1,2,3,4..) are subtrees
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AVL Balancing Four Rotations
X1
X3
Double right
X2
X2
X1
Single right
X3
X1
X1
X3
X2
Single left
Double left
X2
X1
X3
X1
X2
X3
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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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Binary Search Tree for Airport Names
ORY
ZRH
JFK
MEX
BRU
ORD
DUS
ARN
NRT
GLA
GCM
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An AVL Tree for Airport Names
After insertion of ORY, JFK and BRU
Single right
Not AVL balanced
AVL Balanced
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An AVL Tree for Airport Names (contd.)
After insertion of DUS, ZRH, MEX and ORD
After insertion of NRT?
DUS
MEX
ZRH
ORD
Still AVL Balanced
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An AVL Tree
Not AVL Balanaced
Now add ARN and GLA no need for rotations Then
add GCM
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An AVL Tree
JFK
GCM
ARN
DUS
GLA
Double left
NOT AVL BALANCED
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Search Operation

• For successful search, average number of
  comparisons
• sum of all (path length to node1) / number of
  nodes
• For the binary search tree (of airports) it is
• 39/11 3.55
• For the AVL tree (of airports) it is
• 33/11 3.0