1. More Trees
B.Ramamurthy
4/6/2019 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. Complete Binary Tree Is a tree with height h where:
Every node is full upto level h-2,
Level h-1 is completely filled.
Level h is filled from left to right.
Yields to array representation. How to compute indices of parent and child?
How many internal nodes and leafs?
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4. 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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5. 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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6. AVL Property
If N is a node in a binary tree, node N has AVL property if the heights of the left sub-tree and right sub-tree are equal or if they differ by 1.
Lets look at some examples.
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7. AVL Tree: Example
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8. Non-AVL Tree
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9. 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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10. 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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11. Examples for Transformations
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12. 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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13. Binary Search Tree for Airport Names
ORY
JFK
ZRH
MEX
BRU
ORD
DUS
ARN
NRT
GLA
GCM
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14. 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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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+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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20. 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)
AVL trees are no more than 44% worse than optimal trees.
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