The height of record-biased trees

TL;DR

This paper analyzes the height of binary search trees generated from record-biased permutations, revealing it scales with the maximum of two terms involving logarithms, depending on the bias parameter.

Contribution

It provides a complete asymptotic characterization of the height of record-biased binary search trees, extending previous results to a biased permutation model.

Findings

Height scales with max of two logarithmic terms

Results depend on the bias parameter θ

Generalizes classical uniform permutation case

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 $\mathrm{record}(\sigma)=|\{i\in[n]:\forall j<i,\sigma(i)>\sigma(j)\}|$. 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.