Totally bounded ultrametric spaces generated by labeled rays

We will say that a tree T is almost a ray if T is the union of a ray and a finite tree. Let l be a non-degenerate labeling of the vertex set V of almost a ray T and let dI be the corresponding ultrametric on V. It is shown that the ultrametric space (V, dI ) is totally bounded iff this space contains an infinite totally bounded subspace. We also prove that the last property characterizes the almost rays.