The topological trees with extreme matula numbers

Denote by pmthe 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. © 2020 Charles Babbage Research Centre. All rights reserved.