Balanced Binary Trees

We have seen that searching a binary tree is very efficient, O(log(n)); however, this is only true if the tree is balanced, i.e. the two branches of a node have approximately the same height. If the tree is unbalanced, search time could be worse, perhaps even O(n).

There are several clever algorithms that maintain self-balancing binary trees; these algorithms re-balance the tree as needed.

• AVL Tree: heights of subtrees differ by at most 1. A node contains a balance value of -1, 0, or 1, which is the difference in height of its subtrees. If the balance goes out of this range, the tree is rebalanced.

• Red-Black Tree: nodes are colored red or black. The longest path from root to a leaf is no more than twice the length of the shortest path.

• Splay Tree: the tree is rebalanced so that recently accessed elements can be accessed quickly the next time.