Trees
AVL AVL trees are balanced.
An AVL Tree is a binary search tree in which the
heights of siblings can differ by at most 1.
height
0
0
0
0
0
CSE 2011
Prof. J. Elder
0
0
- 16 -
0
0
Last Updated: 4/1/10 2:37 PM
Height of an AVL Tree
Claim: The height of an AVL tree storing n keys is O(log n).
CSE 2011
Prof. J. Elder
- 17 -
Last Updated: 4/1/10 2:37 PM
Insertion
Imbalance may occur at any ancestor of the inserted node.
height = 3
2
1
0
7
Insert(2)
1
4
1
3
0
height = 4
0
5
0
2
0
1
0
2
Problem!
4
1
3
0
1
0
8
0
0
5
0
0
0
CSE 2011
Prof. J. Elder
3
8
7
- 18 -
Last Updated: 4/1/10 2:37 PM
Insertion: Rebalancing Strategy
Step 1: Search
Starting the at inserted node, traverse toward
the root until an imbalance is discovered.
3
2
1
2
height = 4
0
1
Problem!
4
1
3
0
7
8
0
0
5
0
0
CSE 2011
Prof. J. Elder
- 19 -
Last Updated: 4/1/10 2:37 PM
Insertion: Rebalancing Strategy
Step 2: Repair
The repair strategy is called trinode
restructuring.
height = 4
7
3 nodes x, y and z are distinguished:
z = the parent of the high sibling
3
y = the high sibling
x = the high child of the high sibling
2
We can now think of the subtree
rooted at z as consisting of these 3
nodes plus their 4 subtrees
1
2
Problem!
4
1
3
0
1
0
8
0
0
5
0
0
CSE 2011
Prof. J. Elder
- 20 -
Last Updated: 4/1/10 2:37 PM