On the depth of depth-weighted trees

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.