Almost sure asymptotics for the random binary search tree

TL;DR

This paper analyzes the asymptotic behavior of the height and saturation level of a random binary search tree, establishing almost sure and probabilistic results, and investigates the unboundedness of particles at the maximum height.

Contribution

It provides new almost sure and probabilistic asymptotics for the height and saturation level of random binary search trees, including the unboundedness of particles at maximum height.

Findings

Asymptotics for height and saturation level on loglog scale

Almost sure and in probability convergence results

Unbounded number of particles at maximum height

Abstract

We consider a (random permutation model) binary search tree with n nodes and give asymptotics on the loglog scale for the height H_n and saturation level h_n of the tree as n\to\infty, both almost surely and in probability. We then consider the number F_n of particles at level H_n at time n, and show that F_n is unbounded almost surely.