The Height of q-Binary Search Trees

The paper [8] introduces for the first time a meaningful q–model of binary search trees: instead of binary search trees, one considers tournament trees, which differ only marginally from binary search trees; if one starts from a permutation ( 1 2 ··· n π1 π2 ··· πn ) , then one inserts the number i instead of the number πi. Thus, traversing the tree in inorder, we might think of the associated permutation as ρ1σ, where 1 goes into the root, and ρ resp. σ form (recursively) the left resp. right subtree. We could have called this paper “The Height of q-Tournament Trees;” however we decided not to do so since binary search trees are by far better known, both, in the community of theoretical computer scientists, and combinatorialists. A nice reference for tournament trees and increasing trees in general is [2]. Now instead of considering permutations π1π2 · · ·πn, we consider words over the alphabet {1,2,3, . . .}, and (geometric) probabilities attached to the letters, i. e., the probability of letter i is pqi−1, with p + q = 1. The binary search tree is then constructed by writing a nonempty word w as w = xay, where a is the smallest letter occurring, and x ∈ {a+1,a+2, . . .}∗ and y ∈ {a,a+1, . . .}∗. The letter a goes into the root and x resp. y form the left resp. right subtree. The paper [8] dealt with the path length; here we consider the height. The height of a binary search tree (and thus of the trees in our model) is defined to be the largest number of nodes in a path from the root to a leaf; the empty tree (related to the empty word) has height 0. We will prove that the expected height, when considering random words of length n, is asymptotic to pn; the variance will also be computed as well as a Gaussian limit law. (The letter p will always denote 1− q in this paper.) Recall that the result for traditional binary search trees is ∼ c logn with c = 4.31107, see [3]; unfortunately we do not get that result as the limit q→ 1 as it so happened for the path length. However, nothing