The topological trees with extreme Matula numbers

TL;DR

This paper investigates the extremal values of Matula numbers among topological trees, proving that stars minimize and binary caterpillars maximize these numbers for a given number of leaves.

Contribution

It establishes the minimal and maximal Matula numbers for topological trees with fixed leaves, identifying the star and binary caterpillar as extremal cases.

Findings

Stars have the smallest Matula number among all topological trees with fixed leaves.

Binary caterpillars have the largest Matula number among such trees.

Explicit formulas for these extremal values are derived.

Abstract

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