Marco L. Ortiz | Searcharxiv

arXiv2025

A degree-biased cutting process for random recursive trees

We investigate a degree-biased cutting process on random recursive trees, where each vertex is deleted with probability proportional to its degree. We establish the splitting property and derive the explicit distribution of the number of vertices deleted in each cut. This leads to a recursive formula for Kn, the number of cuts needed to erase a random recursive tree with n vertices. Furthermore, we show that Kn is stochastically dominated by Jn, the number of jumps made by a related walk with a barrier. We prove that Jn converges in distribution to a random variable with a spectrally negative stable distribution. Finally, we examine connections between this cutting procedure and a coalescing process on the set of n elements.

math.PR↗

arXiv2022

Targeted cutting of random recursive trees

We propose a method for cutting down a random recursive tree that focuses on its higher degree vertices. Enumerate the vertices of a random recursive tree of size $n$ according to a decreasing order of their degrees; namely, let $(v^{(i)})_{i=1}^{n}$ be so that $deg(v^{(1)}) \geq \cdots \geq deg (v^{(n)})$. The targeted, vertex-cutting process is performed by sequentially removing vertices $v^{(1)}$, $v^{(2)}, \ldots, v^{(n)}$ and keeping only the subtree containing the root after each removal. The algorithm ends when the root is picked to be removed. The total number of steps for this procedure, $X_n^{targ}$, is upper bounded by $Z_{\geq D}$, which denotes the number of vertices that have degree at least as large as the degree of the root. We obtain that the first order growth of $X_n^{targ}$ is upper bounded by $n^{1-\ln 2}$, which is substantially smaller than the required number of removals if, instead, the vertices where selected uniformly at random. More precisely, we prove that $\ln(Z_{\geq D})$ grows as $\ln(n)$ asymptotically and obtain its limiting behavior in probability. Moreover, we obtain that the $k$-th moment of $\ln(Z_{\geq D})$ is proportional to $(\ln(n))^k$.

math.PR↗