The height of Mallows trees

TL;DR

This paper investigates the height of binary search trees built from Mallows permutations, revealing different asymptotic behaviors depending on the parameter q and the relation between n and log n.

Contribution

It extends the understanding of binary search tree heights to Mallows permutations, identifying three regimes of asymptotic behavior based on the parameter q and the size n.

Findings

For q in [0,1], the height asymptotically behaves as (1+o(1))(c* log n + n(1-q)).

Three regimes of height behavior depending on the limit of n(1-q)/log n.

When n(1-q)/log n tends to zero, height is of order c* log n, similar to uniform case.

Abstract

Random binary search trees are obtained by recursively inserting the elements $\sigma(1),\sigma(2),\ldots,\sigma(n)$ of a uniformly random permutation $\sigma$ of $[n]=\{1,\dots,n\}$ into a binary search tree data structure. Devroye (1986) proved that the height of such trees is asymptotically of order $c^*\log n$, where $c^*=4.311\ldots$ is the unique solution of $c \log((2e)/c)=1$ with $c \geq 2$. In this paper, we study the structure of binary search trees $T_{n,q}$ built from Mallows permutations. A $\textrm{Mallows}(q)$ permutation is a random permutation of $[n]=\{1,\ldots,n\}$ whose probability is proportional to $q^{\textrm{Inv}(\sigma)}$, where $\textrm{Inv}(\sigma) = \#\{i < j: \sigma(i) > \sigma(j)\}$. This model generalizes random binary search trees, since $\textrm{Mallows}(q)$ permutations with $q=1$ are uniformly distributed. The laws of $T_{n,q}$ and $T_{n,q^{-1}}$ are related by a simple symmetry (switching the roles of the left and right children), so it suffices to restrict our attention to $q\leq1$. We show that, for $q\in[0,1]$, the height of $T_{n,q}$ is asymptotically $(1+o(1))(c^* \log n + n(1-q))$ in probability. This yields three regimes of behaviour for the height of $T_{n,q}$, depending on whether $n(1-q)/\log n$ tends to zero, tends to infinity, or remains bounded away from zero and infinity. In particular, when $n(1-q)/\log n$ tends to zero, the height of $T_{n,q}$ is asymptotically of order $c^*\log n$, like it is for random binary search trees. Finally, when $n(1-q)/\log n$ tends to infinity, we prove stronger tail bounds and distributional limit theorems for the height of $T_{n,q}$.