Ultrametric spaces generated by labeled star graphs

For an arbitrary star graph S with a non-degenerate vertex labeling l: V (S) → ℝ+, we denote by dl the corresponding ultrametric on the vertex set V (S) of S. We characterize the class US of all ultrametric spaces (V (S), dl) up to isometry. We also find the necessary and sufficient conditions under which the group of all self-isometries of ultrametric space (V (S), dl) coincides with the group of all selfisomorphisms of the labeled star graph S(l).