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 $d(x,c)=\mathrm{diam}B$ whenever $x\in B$ and $x\mathrm{\ne }c$. 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.