1. Tree Balancing: AVL Trees
Dr. Yingwu Zhu

2. Recall in BST The insertion order of items determine the shape of BST
Balanced: search T(n)=O(logN)
Unbalanced: T(n) = O(n)
Key issue:
A need to keep a BST balanced!
Introduce AVL trees, by Russian mathematician

3. AVL Tree Definition First, a BST
Second, height-balance property: balance factor of each node is 0, 1, or -1
Question: what is balance factor?
BF = Height of the left subtree – height of the right subtree
Height: # of levels in a subtree/tree

4. Determine balance factorG F H

5. ADT: AVL Trees Data structure to implement Balance factor Data Left
Right

6. ADT: AVL Trees Basic operations Constructor, search, travesal, empty
Insert: keep balanced!
Delete: keep balanced!
See P842 for class template
Similar to BST

7. Example RI
Insert “DE”, what happens? Need rebalancing? PA

8. Basic Rebalancing Rotation
Single rotation:
Right rotation: the inserted item is on the left subtree of left child of the nearest ancestor with BF of 2
Left rotation: the inserted item is on the right subtree of right child of the nearest ancestor with BF of -2
Double rotation
Left-right rotation: the inserted item is on the right subtree of left child of the nearest ancestor with BF of 2
Right-left rotation: the inserted item is on the left subtree of right child of the nearest ancestor with BF of -2

9. How to perform rotations
Rotations are carried out by resetting links
Two steps:
Determine which rotation
Perform the rotation

10. Right Rotation
Key: identify the nearest ancestor of inserted item with BF +2
A: the nearest ancestor. B: left child
Step1: reset the link from parent of A to B
Step2: set the left link of A equal to the right link of B
Step3: set the right link of B to A
Examples

11. A B
L(A)
L(B)
R(B) h
new

12. Exercise #1 10 4
How about insert 8 20 6 9

13. Left Rotation Step1: reset the link from parent of A to B
Step2: set the right link of A to the left link of B
Step3: set the left link of B to A
Examples

14. Example A B
L(A) h
L(B)
R(B)
new

15. Exercise #2 7 18
What if insert 5 15 10 20

16. Exercise #3 12 1
How about insert 8 16 4 10 14 2 6

17. Basic Rebalancing Rotation
Double rotation
Left-right rotation: the inserted item is on the right subtree of left child of the nearest ancestor with BF of 2
Right-left rotation: the inserted item is on the left subtree of right child of the nearest ancestor with BF of -2

18. Double Rotations Left-right rotation Right-left rotation
How to perform?
1. Rotate child and grandchild nodes of the ancestor
2. Rotate the ancestor and its new child node

19. Example A C
What if insert B

20. Example A B
R(A)
h+1 C
L(B)
h+1
L(C)
R(C) h
new

21. Exercise #4 12 7
How about insert 8 16 4 10 14 2 6

22. Exercise #5 4 2 6 15
What if insert 1 3 5 7 16

23. Exercise #6 4 2 6 14
What if insert 1 3 5 15 7 16

25. Summary on rebalancing
The key is you need to identify the nearest ancestor of inserted item
Determine the rotation by definition
Perform the rotation