The topological trees with extremal Matula numbers

Abstract

Denote by pm the m-th prime number (p1=2, p2=3, p3=5, p4=7, …). Let T be a rooted tree with branches T1,T2,…,Tr. The Matula number M(T) of T is pM(T1)⋅pM(T2)⋅…⋅pM(Tr), starting with M(K1)=1. This number was put forward half a century ago by the American mathematician David Matula. In this paper, we prove that the star (consisting of a root and leaves attached to it) and the binary caterpillar (a binary tree whose internal vertices form a path starting at the root) have the smallest and greatest Matula number, respectively, over all topological trees (rooted trees without vertices of outdegree 1) with a prescribed number of leaves -- the extreme values are also derived.