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∈R+ for which the equation d(x,p)=t has a solution for each p∈X. We prove that the equalities C(X)={0} or C(X)={0,diamX} hold if (X,d) is an ultrametric space generated by labeled trees. The necessary and sufficient conditions under which diamX∈C(X) are found.