A note on the height of binary search trees

Abstract

Let Hn be the height of a binary search tree with n nodes constructed by standard insertions from a random permutation of 1, …, n. It is shown that Hn/log n → c = 4.31107 … in probability as n → ∞, where c is the unique solution of c log((2e)/c) = 1, c ≥ 2. Also, for all p > 0, limn→∞E(Hpn)/ logpn = cp. Finally, it is proved that Sn/log n → c* = 0.3733 …, in probability, where c* is defined by c log((2e)/c) = 1, c ≤ 1, and Sn is the saturation level of the same tree, that is, the number of full levels in the tree. © 1986, ACM. All rights reserved.