The height of record-biased trees

Abstract

Given a permutation $\sigma$, its corresponding binary search tree is obtained by recursively inserting the values $\sigma(1),\ldots,\sigma(n)$ into a binary tree so that the label of each node is larger than the labels of its left subtree and smaller than the labels of its right subtree. In 1986, Devroye proved that the height of such trees when $\sigma$ is a random uniform permutation is of order $(c^*+o_\mathbb{P}(1))\log n$ as $n$ tends to infinity, where $c^*$ is the only solution to $c\log(2e/c)=1$ with $c\geq2$. In this paper, we study the height of binary search trees drawn from the record-biased model of permutations, introduced by Auger, Bouvel, Nicaud, and Pivoteau in 2016. The record-biased distribution is the probability measure on the set of permutations whose weight is proportional to $\theta^{\mathrm{record}(\sigma)}$, where $\mathrm{record}(\sigma)=|\{i\in[n]:\forall j \sigma(j)\}|$. We show that the height of a binary search tree built from a record-biased permutation of size $n$ with parameter $\theta$ is of order $(1+o_\mathbb{P}(1))\max\{c^*\log n,\,\theta\log(1+n/\theta)\}$, hence giving a full characterization of the first order asymptotic behaviour of the height of such trees.