1. AVL Tree
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2. Outline
AVL Tree
Definition
Properties
Operations
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3. AVL Trees
Unbalanced Binary Search Trees are bad. Worst case: operations take O(n).
AVL (Adelson-Velskii & Landis) trees maintain balance.
For each node in tree, height of left subtree and height of right subtree differ by a maximum of 1. X X H
H-2
H-1
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4. AVL Trees 10 5 3 20 1 43 10 5 3 20 5 2 1 3
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5. AVL Trees 12 8 16 4 10 14 2 6
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6. Insertion for AVL Tree After insert 1 12 8 16 4 10 14 2 6 1
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7. Insertion for AVL Tree
To ensure balance condition for AVL-tree, after insertion of a new node, we back up the path from the inserted node to root and check the balance condition for each node.
If after insertion, the balance condition does not hold in a certain node, we do one of the following rotations:
Single rotation
Double rotation
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8. Insertions Causing Imbalance
HP=HQ=HR k2 k1 k1 k2 R P P Q Q R
An insertion into the subtree:
P (outside) - case 1
Q (inside) - case 2
An insertion into the subtree:
Q (inside) - case 3
R (outside) - case 4
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9. Single Rotation (case 1)
HA=HB+1
HB=HC k2 k1 k1 k2 A C A B B C
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10. Single Rotation (case 4)
HA=HB
HC=HB+1 k2 k1 k1 k2 C A A B B C
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11. Problem with Single Rotation
Single rotation does not work for case 2 and 3 (inside case) k2 k1 k1 k2 R P P Q Q R
HQ=HP+1
HP=HR
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12. Double Rotation: Step k3 k3 k1 k2 D D k2 k1 A C B C B A HA=HB=HC=HD
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13. Double Rotation: Step k2 k1 k3 A D B C
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14. Double Rotation k3 k2 k1 k1 k3 D k2 A B C D A B C HA=HB=HC=HD
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15. Double Rotation k2 k1 k1 k3 k3 A k2 A B C D D B C HA=HB=HC=HD
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16. Example Insert 3 into the AVL tree 11 8 20 4 16 27 8 11 4 20 3 16 27 8
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17. Example Insert 5 into the AVL tree 11 8 20 4 16 27 8 11 5 20 4 16 27 8
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18. AVL Trees: Exercise Insertion order:
10, 85, 15, 70, 20, 60, 30, 50, 65, 80, 90, 40, 5, 55
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19. Remove Operation in AVL Tree
Removing a node from an AVL Tree is the same as removing from a binary search tree. However, it may unbalance the tree.
Similar to insertion, starting from the removed node we check all the nodes in the path up to the root for the first unbalance node.
Use the appropriate single or double rotation to balance the tree.
May need to continue searching for unbalanced nodes all the way to the root.
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20. Deletion X in AVL Trees Deletion: Case 1: if X is a leaf, delete X
Case 2: if X has 1 child, use it to replace X
Case 3: if X has 2 children, replace X with its inorder predecessor (and recursively delete it)
Rebalancing
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21. Delete 55 (case 1) 60 20 70 10 40 65 85 80 90 5 15 30 50 55
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22. Delete 55 (case 1) 60 20 70 10 40 65 85 80 90 5 15 30 50 55
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23. Delete 50 (case 2) 60 20 70 10 40 65 85 80 90 5 15 30 50 55
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24. Delete 50 (case 2) 60 20 70 10 40 65 85 50 80 90 5 15 30 55
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25. Delete 60 (case 3) 60
prev 20 70 10 40 65 85 80 90 5 15 30 50 55
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26. Delete 60 (case 3) 55 20 70 10 40 65 85 80 90 5 15 30 50
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27. Delete 55 (case 3) 55
prev 20 70 10 40 65 85 80 90 5 15 30 50
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28. Delete 55 (case 3) 50 20 70 10 40 65 85 80 90 5 15 30
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29. Delete 50 (case 3) 50
prev 20 70 10 40 65 85 80 90 5 15 30
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30. Delete 50 (case 3) 40 20 70 10 30 65 85 80 90 5 15
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31. Delete 40 (case 3) 40
prev 20 70 10 30 65 85 80 90 5 15
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32. Delete 40 : Rebalancing 30 20 70 10 Case ? 65 85 80 90 5 15
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33. Delete 40: after rebalancing30 10 70 20 65 85 5 80 90 15
Single rotation is preferred!
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34. Minimum Element in AVL Tree
An AVL Tree of height H has at least FH+3-1 nodes, where Fi is the i-th fibonacci number
S0 = 1
S1 = 2
SH = SH-1 + SH-2 + 1
SH-1
SH-2 H
H-2
H-1
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35. AVL Tree: analysis (1)
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36. AVL Tree: analysis (2) The depth of AVL Trees is at most logarithmic.
So, all of the operations on AVL trees are also logarithmic.
The worst-case height is at most 44 percent more than the minimum possible for binary trees.
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37. Summary
Find element, insert element, and remove element operations all have complexity O(log n) for worst case
Insert operation: top-down insertion and bottom up balancing
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38. Further Study
Section 19.4 of Weiss book
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