Some average performance measures for the B-tree

Summary

Several average performance measures are presented for large B-trees formed from insertions, where large refers to the number of keys. Formulas are first derived for the expected number of nodes of each size on the bottom level of the tree, where the size of a node is the number of keys currently contained in the node. This is followed by formulas for the probability of making an insertion into a node of a given size, the probability of a split during an insertion into a node, and the expected number of splits during an insertion into the tree. It is shown that for large trees of high order m, the expected number of splits per insertion is approximately 1/((ln 2) m). A formula is presented for the average storage utilization, and it is shown that this average approaches ln 2 as m approaches infinity. A simpler formula is derived for the average storage utilization at the bottom level of the tree, and it is shown that this formula is an increasing function of m ranging from 2/3 to ln 2. It is shown that the expected tree height and the expected search path length are approximately logarithmic to the base (ln 2) m. Simulation results are presented to corroborate the theoretical analysis.