Totally bounded ultrametric spaces generated by labeled rays

We will say that an infinite 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 d_l be the corresponding ultrametric on V.  It is shown that the ultrametric space (V, d_l) is totally bounded if this space contains an infinite totally bounded subspace.  We also prove that the last property characterizes the almost rays.