Oscillations in the height of the Yule tree and application to the binary search tree

TL;DR

This paper proves that the height of the Yule tree exhibits oscillations around a critical distribution and applies these findings to the height and saturation level of binary search trees.

Contribution

It establishes the oscillatory behavior of the Yule tree's height distribution and extends these results to binary search trees, revealing new insights into their asymptotic properties.

Findings

Distribution of the centered maximum oscillates around a critical wave

Oscillations occur for expectations of functions of the height

Results apply to binary search tree height and saturation level

Abstract

For a particular case of a branching random walk with lattice support, namely the Yule branching random walk, we prove that the distribution of the centred maximum oscillates around a distribution corresponding to a critical travelling wave in the following sense: there exist continuous functions $t \mapsto a_t$ and $x \mapsto \overline{\phi}(x)$ such that: $$\lim_{t \rightarrow +\infty} \sup_{x \in \mathbb{R}} \vert \mathbb{P}(\overline{X}(t) \leq a_t +x )-\overline{\phi}(x- \{ a_t +x\})\vert=0,$$ where $\{x\}=x-\lfloor x \rfloor$ and $\overline{X}(t)$ is the height of the Yule tree. We also shows that similar oscillations occur for $\mathbb{E}\left(f(\overline{X}(t)-a_t)\right)$, when $f$ is in a large class of functions. This process is classically related to the binary search tree, thus yielding analogous results for the height and for the saturation level of the binary search tree.