Expected number of induced subtrees shared by two independent copies of the terminal tree in a critical branching process

Boris Pittel

Published 2021-02-11Version 1

Consider a rooted tree $T$ of minimum degree $2$ at least, with leaf-set $[n]$. A rooted tree $\mathcal T$ with leaf-set $S\subset [n]$ is induced by $S$ in $T$ if $\mathcal T$ is the lowest common ancestor subtree for $S$, with all its degree-2 vertices suppressed. A "maximum agreement subtree" (MAST) for a pair of two trees $T'$ and $T''$ is a tree $\mathcal T$ with a largest leaf-set $S\subset [n]$ such that $\mathcal T$ is induced by $S$ both in $T'$ and $T''$. Bryant et al. \cite{BryMcKSte} and Bernstein et al. \cite{Ber} proved, among other results, that for $T'$ and $T''$ being two independent copies of a random binary (uniform or Yule-Harding distributed) tree $T$, the likely magnitude order of $\text{MAST}(T',T'')$ is $O(n^{1/2})$. In this paper we prove this bound for a wide class of random rooted trees: $T$ is a terminal tree of a branching process with an offspring distribution of mean $1$, conditioned on "total number of leaves is $n$".