Locally finite ultrametric spaces and labeled trees

It is shown that a locally finite ultrametric space (X, d) is generated by labeled tree if and only if, for every open ball B ⊆ X, there is a point c ∈ B such that d(x, c) = diam B whenever x ∈ B and x 6= c. For every finite ultrametric space Y we construct an ultrametric space Z having the smallest possible number of points such that Z is generated by labeled tree and Y is isometric to a subspace of Z. It is proved that for a given Y , such a space Z is unique up to isometry.