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 iff this space contains an infinite totally bounded subspace. We also prove that the last property characterizes the almost rays.