On the peel number and the leafheight of a GaltonWatson tree
Abstract
We study several parameters of a random BienayméGaltonWatson tree $T_n$ of size $n$ defined in terms of an offspring distribution $\xi$ with mean $1$ and nonzero finite variance $\sigma^2$. Let $f(s)={\bf E}\{s^\xi\}$ be the generating function of the random variable $\xi$. We show that the independence number is in probability asymptotic to $qn$, where $q$ is the unique solution to $q = f(1q)$. One of the many algorithms for finding the largest independent set of nodes uses a notion of repeated peeling away of all leaves and their parents. The number of rounds of peeling is shown to be in probability asymptotic to $\log n / \log\bigl(1/f'(1q)\bigr)$. Finally, we study a related parameter which we call the leafheight. Also sometimes called the protection number, this is the maximal shortest path length between any node and a leaf in its subtree. If $p_1 = {\bf P}\{\xi=1\}>0$, then we show that the maximum leafheight over all nodes in $T_n$ is in probability asymptotic to $\log n/\log(1/p_1)$. If $p_1 = 0$ and $\kappa$ is the first integer $i>1$ with ${\bf P}\{\xi=i\}>0$, then the leafheight is in probability asymptotic to $\log_\kappa\log n$.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.14389
 Bibcode:
 2021arXiv210614389D
 Keywords:

 Mathematics  Probability;
 60J80
 EPrint:
 20 pages, 5 figures, 1 table, revised according to referee suggestions, added missing "the" in the arXiv metadata title