Distribution of distances in random binary search trees

Abstract

We investigate random distances in a random binary search tree. Two types of random distance are considered: the depth of a node randomly selected from the tree, and distance between randomly selected pairs of nodes. By a combination of classical methods and modern contraction techniques we arrive at a Gaussian limit law for normed random distances between pairs. The exact forms of the mean and variance of this latter distance are first derived by classical methods to determine the scaling properties, then used for norming, and the normed random variable is then shown by the contraction method to have a normal limit arising as the fixed-point solution of a distributional equation. We identify the rate of convergence in the limit law to be of the order $\Theta(1/\sqrt{\ln n})$ in the Zolotarev metric $\zeta_3$. In the analysis we need the rate of convergence in the central limit law for the depth of a node, as well. This limit law was derived before by various techniques. We establish the rate $\Theta(1/\sqrt{\ln n})$ in $\zeta_3$.