1. AVL Trees
CSE 373
Data Structures

2. Binary Search Tree - Best Time
All BST operations are O(d), where d is tree depth
minimum d is for a binary tree with N nodes
What is the best case tree?
What is the worst case tree?
So, best case running time of BST operations is O(log N)
1/15/2019
CSE AU AVL Trees

3. Binary Search Tree - Worst Time
Worst case running time is O(N)
What happens when you Insert elements in ascending order?
Insert: 2, 4, 6, 8, 10, 12 into an empty BST
Problem: Lack of “balance”:
compare depths of left and right subtree
Unbalanced degenerate tree
1/15/2019
CSE AU AVL Trees

4. Balanced and unbalanced BST1 4 2 2 5 3 1 3 4 4
Is this “balanced”? 5 2 6 6 1 3 5 7 7
1/15/2019
CSE AU AVL Trees

5. Approaches to balancing trees
Don't balance
May end up with some nodes very deep
Strict balance
The tree must always be balanced perfectly
Pretty good balance
Only allow a little out of balance
Adjust on access
Self-adjusting
1/15/2019
CSE AU AVL Trees

6. Balancing Binary Search Trees
Many algorithms exist for keeping binary search trees balanced
Adelson-Velskii and Landis (AVL) trees (height-balanced trees)
Splay trees and other self-adjusting trees
B-trees and other multiway search trees
1/15/2019
CSE AU AVL Trees

7. Perfect Balance Want a complete tree after every operation
tree is full except possibly in the lower right
This is expensive
For example, insert 2 in the tree on the left and then rebuild as a complete tree 6 5
Insert 2 &
complete tree 4 9 2 8 1 5 8 1 4 6 9
1/15/2019
CSE AU AVL Trees

8. AVL - Good but not Perfect Balance
AVL trees are height-balanced binary search trees
Balance factor of a node
height(left subtree) - height(right subtree)
An AVL tree has balance factor calculated at every node
For every node, heights of left and right subtree can differ by no more than 1
Store current heights in each node
1/15/2019
CSE AU AVL Trees

9. Height of an AVL Tree
N(h) = minimum number of nodes in an AVL tree of height h.
Basis
N(0) = 1, N(1) = 2
Induction
N(h) = N(h-1) + N(h-2) + 1
Solution (Fibonacci)
N(h) > h (  1.62) h
h-2
h-1
1/15/2019
CSE AU AVL Trees

10. Height of an AVL Tree N(h) > h (  1.62)
Suppose we have n nodes in an AVL tree of height h.
n > N(h) (because N(h) was the minimum)
n > h hence log n > h (relatively well balanced tree!!)
h < 1.44 log2n (i.e., O(height) takes O(logn))
1/15/2019
CSE AU AVL Trees

11. Node Heights 6 6 4 9 4 9 1 5 1 5 8 Tree A (AVL) Tree B (AVL)
height=3 balance factor=2-1=1 3 6 6 2 1 2 2 4 9 4 9 1 1 1 1 1 1 5 1 5 8
1/15/2019
CSE AU AVL Trees

12. Worst-case AVL Trees 4 nodes: h = 3 vs 3 7 nodes: h = 4 vs 3
1/15/2019
CSE AU AVL Trees

13. Node Heights after Insert 7
Tree A (AVL)
Tree B (not AVL)
balance factor
2-0 = 2 3 4 6 6 2 2 2 3 4 9 4 9 1 1 1 1 1 2 1 5 7 1 5 8 1 7
1/15/2019
CSE AU AVL Trees

14. Insert and Rotation in AVL Trees
Insert operation may cause balance factor to become 2 or –2 for some node
only nodes on the path from insertion point to root node have possibly changed in height
So after the Insert, go back up to the root node by node, updating heights
If a new balance factor (the difference hleft-hright) is 2 or –2, adjust tree by rotation around the node
1/15/2019
CSE AU AVL Trees

15. Single Rotation in an AVL Tree3 3 6 6 2 3 2 2 4 9 4 8 1 1 1 2 1 1 1 1 5 8 1 5 7 9 1 7
1/15/2019
CSE AU AVL Trees

16. Insertions in AVL Trees
Let the node that needs rebalancing be .
There are 4 cases:
Outside Cases (require single rotation) :
1. Insertion into left subtree of left child of .
2. Insertion into right subtree of right child of .
Inside Cases (require double rotation) :
3. Insertion into right subtree of left child of .
4. Insertion into left subtree of right child of .
The rebalancing is performed through four separate rotation algorithms.
1/15/2019
CSE AU AVL Trees

17. AVL Insertion: Outside Casef
Consider a valid
AVL subtree b h G h h A D
1/15/2019
CSE AU AVL Trees

18. AVL Insertion: Outside Casef
Inserting into A
destroys the AVL
property at node f b h G
h+1 h D A
1/15/2019
CSE AU AVL Trees

19. AVL Insertion: Outside Casef
Do a “right rotation” b h G
h+1 h D A
1/15/2019
CSE AU AVL Trees

20. f b G D A Single right rotation Do a “right rotation” h h+1 h
1/15/2019
CSE AU AVL Trees

21. Outside Case Completedb
“Right rotation” done!
(“Left rotation” is mirror
symmetric) f
h+1 h h A G D
AVL property has been restored!
1/15/2019
CSE AU AVL Trees

22. AVL Insertion: Inside Casef
Consider a valid
AVL subtree b h G h h A D
1/15/2019
CSE AU AVL Trees

23. AVL Insertion: Inside Casef
Inserting into D
destroys the
AVL property
at node f
Does “right rotation”
restore balance? b h G h
h+1 A D
1/15/2019
CSE AU AVL Trees

24. AVL Insertion: Inside Caseb
“Right rotation”
does not restore
balance… now b is
out of balance f h A h
h+1 G D
1/15/2019
CSE AU AVL Trees

25. AVL Insertion: Inside Casef
Consider the structure
of subtree Y… b h G h
h+1 A D
1/15/2019
CSE AU AVL Trees

26. AVL Insertion: Inside Casef
Y = node i and
subtrees V and W b h G d h
h+1 A
h or h-1 C E
1/15/2019
CSE AU AVL Trees

27. AVL Insertion: Inside Casef
We will do a left-right
“double rotation” . . . b G d A C E
1/15/2019
CSE AU AVL Trees

28. Double rotation : first rotation
left rotation complete d G b E C A
1/15/2019
CSE AU AVL Trees

29. Double rotation : second rotationf
Now do a right rotation d G b E C A
1/15/2019
CSE AU AVL Trees

30. Double rotation : second rotation
right rotation complete
Balance has been
restored d b f h h
h or h-1 C A E G
1/15/2019
CSE AU AVL Trees

31. Example of Insertions in an AVL Tree3 20
Insert 5, 40 1 2 10 30 1 1 25 35
1/15/2019
CSE AU AVL Trees

32. Example of Insertions in an AVL Tree3 4 20 20 2 2 2 3 10 30 10 30 1 1 1 2 1 1 5 25 35 5 25 35 1 40
Now Insert 45
1/15/2019
CSE AU AVL Trees

33. Single rotation (outside case)5 4 20 20 2 4 2 3 10 30 10 30 1 3 1 1 1 2 5 25 35 5 25 40 1 1 35 45 40 2
Imbalance 1 45
Now Insert 34
1/15/2019
CSE AU AVL Trees

34. Double rotation (inside case)5 4 20 20 4 4 2 3 10 30 10 35 1 3 3 1 2 2 5
Imbalance 25 40 5 30 40 1 25 34 2 1 45 35 45 1
Insertion of 34 1 34
1/15/2019
CSE AU AVL Trees

35. AVL Tree Deletion Similar but more complex than insertion
Rotations and double rotations needed to rebalance
Imbalance may propagate upward so that many rotations may be needed.
1/15/2019
CSE AU AVL Trees

36. Pros and Cons of AVL Trees
Arguments for AVL trees:
Search is O(log N) since AVL trees are always balanced.
Insertion and deletions are also O(logn)
The height balancing adds no more than a constant factor to the speed of insertion.
Arguments against using AVL trees:
Difficult to program & debug; more space for balance factor.
Asymptotically faster but rebalancing costs time.
Most large searches are done in database systems on disk and use other structures (e.g. B-trees).
May be OK to have O(N) for a single operation if total run time for many consecutive operations is fast (e.g. Splay trees).
1/15/2019
CSE AU AVL Trees

37. Deletion 6
100 4 5 44
144 2 3 4 3 17 62
117
162 1 1 2 2 3 2 2 32 50 78
111
132
150
178 1 1 2 1 1 1 1 72
101
128
132
147
172
181 1
129
1/15/2019
CSE AU AVL Trees

38. Deletion 6
100 4 5 44
144 1 3 4 3 17 62
117
162 1 1 2 2 3 2 2 32 50 78
111
132
150
178 1 1 2 1 1 1 1 72
101
128
132
147
172
181 1
129
1/15/2019
CSE AU AVL Trees

39. Deletion 6
100 5 44
144 4 3 17 62
117
162 2 3 2 2 50 78
111
132
150
178 1 2 1 1 1 1 72
101
128
132
147
172
181 1
129
1/15/2019
CSE AU AVL Trees

40. Deletion 6
100 5 62
144 4 3 44
117
162 78 2 3 2 2 17 50 72
111
132
150
178 1 2 1 1 1 1
101
128
132
147
172
181 1
129
1/15/2019
CSE AU AVL Trees

41. Deletion 6
100 3 5 62
144 2 4 3 2 44
117
162 78 1 1 1 2 3 2 2 17 50 72
111
132
150
178 1 2 1 1 1 1
101
128
132
147
172
181 1
129
1/15/2019
CSE AU AVL Trees

42. Deletion 6
100 3 5 62
144 2 4 3 2 44
117
162 78 1 1 1 2 3 2 2 17 50 72
111
132
150
178 1 2 1 1 1 1
101
128
132
147
172
181 1
129
1/15/2019
CSE AU AVL Trees

43. Deletion
100 62
144 44
117
162 78 17 50 72
111
132
150
178
101
128
132
147
172
181
129
1/15/2019
CSE AU AVL Trees

44. Deletion
100 62
117 44
144 78
111 17 50 72
101
132
162
128
132
150
178
129
147
172
181
1/15/2019
CSE AU AVL Trees

45. Deletion
117
100
144 62
111
132
162 44 78
101
128
132
150
178 17 50 72
129
147
172
181
1/15/2019
CSE AU AVL Trees

46. Deletion 5
117 4 4
100
144 3 2 3 3 62
111
132
162 2 2 1 2 1 2 2 44 78
101
128
132
150
178 1 1 1 1 1 1 1 17 50 72
129
147
172
181
1/15/2019
CSE AU AVL Trees