In the article cited below, I. Castellano gives a proof for the following result (Proposition 4.1).
Let $G$ be a compactly generated totally disconnected locally compact group. Suppose that $G$ acts discretely on a tree $\mathcal T$ such that
- the group $G$ is acting without edge inversions;
- the quotient graph $G\backslash\mathcal T$ is finite;
- the edge stabilisers $G_e$ are compact open subgroups of $G$.
Then the vertex stabilizers $G_v$ are compactly generated.
Here acting discretely means that the stabilizers are open subgroups of $G$.
The proof as cited is rather involved and uses cohomology arguments, while the proposition looks rather innocent. Is there a more straightforward, elementary proof for this result? Note that the proposition should also hold for trees that are not locally finite (and in this case, the topology becomes a lot more subtle).
 Castellano, I., Rational discrete first degree cohomology for totally disconnected locally compact groups. arXiv:1506.02310 [math.GR].