The topological trees with extreme Matula numbers

Abstract

Denote by $p_m$ the $m$-th prime number ($p_1=2$, $p_2=3$, $p_3=5$, $p_4=7$, \dots). Let $T$ be a rooted tree with branches $T_1,T_2,\dots ,T_r$. The Matula number $M(T)$ of $T$ is $p_{M(T_1)}\cdot p_{M(T_2)}\cdots p_{M(T_r)}$, starting with $M(K_1)=1$. 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.