Balanced Binary Search Trees

1
Balanced Binary Search Trees

• height is O(log n), where n is the number of
  elements in the tree
• AVL (Adelson-Velsky and Landis) trees
• red-black trees
• find, insert, and erase take O(log n) time

2
Balanced Binary Search Trees

• Indexed AVL trees
• Indexed red-black trees
• Indexed operations also take
• O(log n) time

3
Balanced Search Trees

• weight balanced binary search trees
• 2-3 2-3-4 trees
• AA trees
• B-trees
• BBST
• etc.

4
AVL Tree

• binary tree
• for every node x, define its balance factor
• balance factor of x height of left subtree of x
  
• - height of
  right subtree of x
• balance factor of every node x is -1, 0, or 1

5
Balance Factors
-1
1
1
-1
0
1
0
0
-1
0
0
0
0

• This is an AVL tree.

6
Height

• The height of an AVL tree that has n nodes is at
  most 1.44 log2 (n2).
• The height of every n node binary tree is at
  least log2 (n1).

7
AVL Search Tree
8
insert(9)
-1
10
0
1
1
7
40
-1
0
1
-1
0
45
8
3
30
0
-1
0
0
0
0
60
35
9
1
20
5
0
25
9
insert(29)
-1
10
1
1
7
40
-1
0
1
0
45
8
3
30
0
-2
0
-1
0
0
60
35
1
20
5
0
-1
RR imbalance gt new node is in right subtree of
right subtree of blue node (node with bf -2)
25
0
29
10
insert(29)
-1
10
1
1
7
40
-1
0
1
0
45
8
3
30
0
0
0
0
0
60
35
1
25
5
0
0
20
29
RR rotation.