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.