GNU libavl 2.0.1
Author: Ben Pfaff

5.1.1 Analysis |

How good is the AVL balancing rule? That is, before we consider how much complication it adds to BST operations, what does this balancing rule guarantee about performance? This is a simple question only if you're familiar with the mathematics behind computer science. For our purposes, it suffices to state the results:

> An AVL tree with n nodes has height between log2 (n + 1) and 1.44 * log2 (n + 2) - 0.328. An AVL tree with height h has between pow (2, (h + .328) / 1.44) - 2 and pow (2, h) - 1 nodes.

> For comparison, an optimally balanced BST with n nodes has height ceil (log2 (n + 1)). An optimally balanced BST with height h has between pow (2, h - 1) and pow (2, h) - 1 nodes.1

The average speed of a search in a binary tree depends on the tree's height, so the results above are quite encouraging: an AVL tree will never be more than about 50% taller than the corresponding optimally balanced tree. Thus, we have a guarantee of good performance even in the worst case, and optimal performance in the best case.

See also: [Knuth 1998b], theorem 6.2.3A.

#### Footnotes

1. Here log2 is the standard C base-2 logarithm function, pow is the exponentiation function, and ceil is the ``ceiling'' or ``round up'' function. For more information, consult a C reference guide, such as [Kernighan 1988].

5.1 Balancing Rule | 5.1 Balancing Rule | 5.2 Data Types