On the depth of depth-weighted trees

TL;DR

This paper provides a detailed analysis of the depth of depth-weighted trees grown from a root, offering precise formulas for various weight functions and confirming a conjecture for certain growth types.

Contribution

It systematically characterizes the typical depth of depth-weighted trees for different classes of weight functions, including convergent, periodic, and exponential growth.

Findings

Precise expressions for typical depth across various weight functions

Confirmation of a conjecture for bounded and exponential weights

Strengthening previous conjectures with rigorous analysis

Abstract

The depth-weighted tree DWT($f$) with weight function $f:\{0,1,2,\ldots\}\to (0,\infty)$ is a dynamic random tree grown from a root $r$ where vertices arrive consecutively and every new vertex attaches to a parent $u$ with probability proportional to $f$(distance between $u$ and $r$). This work is dedicated to a systematic analysis of the depth of DWT($f$). Namely, we provide precise analytic expressions of the typical depth of DWT($f$) for convergent, periodic, slowly growing, and (super-)exponentially growing weight functions. Furthermore, for bounded or exponentially growing $f$, we determine the typical depth up to a multiplicative constant, thus confirming and strengthening a conjecture of Leckey, Mitsche and Wormald.