I read the paper "Kurosh rank of intersections of subgroups of free products of orderable groups", which can be found here: http://arxiv.org/abs/1109.0233. The proposition 2.3 states that: Let $H$ be a subgroup of $G$. Let $T'$ be a $H$-subtree of $T$. Then $κ_{T}(H)$ = $κ_{T′}(H)$. In particular, $T_{H}/H$ is finite (as a graph) if and only if $κ_T(H)$ is finite.
They leave it as Bass-Serre exercise. I am not very good in Bass-Serre theory, so I would appreciate if anyone can help me with this proof.