Spanning Tree Size in Random Binary Search Trees

This paper deals with the size of the spanning tree of p randomly chosen nodes in a binary search tree. It is shown via generating functions methods, that for fixed p, the (normalized) spanning tree size converges in law to the Normal distribution. The special case p = 2 reproves the recent result (obtained by the contraction method by Mahmoud and Neininger [Ann. Appl. Probab. 13 (2003) 253–276]), that the distribution of distances in random binary search trees has a Gaussian limit law. In the proof we use the fact that the spanning tree size is closely related to the number of passes in Multiple Quickselect. This parameter, in particular, its first two moments, was studied earlier by Panholzer and Prodinger [Random Structures Algorithms 13 (1998) 189–209]. Here we show also that this normalized parameter has for fixed p-order statistics a Gaussian limit law. For p = 1 this gives the well-known result that the depth of a randomly selected node in a random binary search tree converges in law to the Normal distribution. 1. Introduction. In the papers [7] and [1] the distances between nodes in random search trees, respectively, random recursive trees were studied. It was proven in [7] that the (edge) distances ∆ n between two randomly selected nodes in random binary search trees of size n are asymptotically normally (Gaussian) distributed, where the so-called random permutation model was used as the model of randomness for the trees. This means that every permutation of length n is assumed to be equally likely when generating a binary search tree; furthermore, for selecting nodes, all