(Un)expected path lengths of asymmetric binary search trees

Abstract

The average path length over the family of binary search trees with 2r+1 nodes, built from randomly choosen permutations of r distinct keys, is O(r log(r)). On the other hand, if the probability that the r keys are already sorted, tends towards 1, the resulting family of binary search trees degenerates, until it consists of the single linear list - tree with a worst case path length of r(r + 1). Subject of the paper is the gap between the orders of growth of the expected path lengths of these two models. We systematically deform the probability model of binary search trees using three different kinds of deformers which are controlled by a real constant c and, using singularity analysis, we obtain the surprising result, that under two deformations the expected path length of a tree of size 2r+1 is either O(r log(r)) or O(r2).The same result is valid under the third deformation for special values of c.