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 ${ \bf US}$ denote the class of all ultrametric spaces generated by vertex labelings of star graphs. We prove that the inclusion $U(T)\subseteq { \bf US}$ holds if and only if the longest path in $T$ has a length not exceeding three.
Categories: math.GN