Expected number of induced subtrees shared by two independent copies of a random tree

Abstract

Consider a rooted tree T of minimum degree 2 at least, with leaf-set [n]. A rooted tree T with leaf-set S ⊂ [n] is induced by S in T if 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 T with a largest leaf-set S ⊂ [n] such that T is induced by S both in T ′ and T ′′ . Bryant et al. [7] and Bernstein et al.[6] 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 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".