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 $B \subseteq X$, there is a point $c \in B$ such that $d(x, c) = \operatorname{diam} B$ whenever $x \in B$ and $x \neq c$. 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.