$k$-protected vertices in binary search trees

TL;DR

This paper investigates the probability distribution of vertices at a certain distance from leaves in large random binary search trees, revealing convergence to specific rational constants.

Contribution

It establishes the asymptotic probability that a vertex in a random binary search tree is at a fixed distance from the nearest leaf, for all k.

Findings

Probability converges to rational constants as n increases.

Explicit characterization of the distribution of k-protected vertices.

Provides a new understanding of the structure of large binary search trees.

Abstract

We show that for every $k$, the probability that a randomly selected vertex of a random binary search tree on $n$ nodes is at distance $k-1$ from the closest leaf converges to a rational constant $c_k$ as $n$ goes to infinity.