1. AVL Trees
B.Ramamurthy
4/27/2019 BR

2. 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
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3. 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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4. 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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5. Unbalanced Tree 45 54 65
Maximum height 78 87 98
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6. Balanced Tree 78 87 87 98 87 87
Minimum Height
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7. 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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8. AVL Tree: Example
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9. Non-AVL Tree
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10. 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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11. 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 B A B A B C A C B A B C A B C
A,B,C are nodes, Tn (n =1,2,3,4..) are subtrees
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12. AVL Balancing : Four RotationsX3 X2 X1 X1 X3
Double right X2 X3 X1 X2
Single right X2 X1 X3 X1 X2 X3 X1 X3 X2
Single left
Double left X2 X1 X3 X1 X2 X3
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13. 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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14. Binary Search Tree for Airport Names
ORY
JFK
ZRH
MEX
BRU
ORD
DUS
ARN
NRT
GLA
GCM
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15. An AVL Tree for Airport Names
After insertion of ORY, JFK and BRU :
ORY
JFK
BRU
Single right
JFK
BRU
ORY
Not AVL balanced
AVL Balanced
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16. An AVL Tree for Airport Names (contd.)
After insertion of DUS, ZRH,
MEX and ORD
After insertion of NRT?
JFK
BRU
ORY
DUS
MEX
ZRH
ORD
NRT
JFK
BRU
ORY
DUS
MEX
ZRH
ORD
Still AVL Balanced
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17. An AVL Tree … Not AVL Balanaced Double Left Now add ARN and GLA; no
JFK
BRU
ORY
DUS
MEX
ZRH
ORD
NRT
JFK
BRU
ORY
DUS
NRT
ZRH
ORD
MEX
Double Left
Now add ARN and GLA; no
need for rotations; Then add GCM
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18. An AVL Tree… Double left JFK BRU DUS ORY NRT ZRH ORD MEX ARN GLA GCM
NOT AVL BALANCED
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19. Search Operation For successful search, average number of comparisons:
sum of all (path length to node+1) / 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
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