Center of distances of ultrametric spaces generated by labeled trees

The center of distances of a metric space $(X,d)$ is the set $C(X)$ of all $t\in \mathbb R^+$ for which the equation $d(x,p)=t$ has a solution for each $p\in X$. We prove that the equalities $C(X)=\{0\}$ or $C(X)=\{0,\operatorname{diam}X\}$ hold if $(X,d)$ is an ultrametric space generated by labeled trees. The necessary and sufficient conditions under which $\operatorname{diam} X\in C(X)$ are found.