Locally finite ultrametric spaces and labeled trees

It is shown that a locally finite ultrametric space ( X , d ) is generated by a 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 ≠ 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 a 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.