Longest paths in trees and isometricity of ultrametric spaces

Abstract

Let $T$ be a tree of arbitrary finite or infinite order and let $U(T)$ be the set of all ultrametric spaces generated by vertex labelings of $T$. Let $\mathbf{US}$ denote the class of all ultrametric spaces generated by vertex labelings of star graphs. We prove that the inclusion $U(T)\subseteq \mathbf{US}$ holds if and only if the longest path in $T$ has a length not exceeding three.