  Let $T$ be a locally finite tree without vertices of degree $1$. We show that
among the closed subgroups of $\mathrm{Aut}(T)$ acting with a bounded number of
orbits, the Chabauty-closure of the set of topologically simple groups is the
set of groups without proper open subgroup of finite index. Moreover, if all
vertices of $T$ have degree $\geq 3$, then the set of isomorphism classes of
topologically simple closed subgroups of $\mathrm{Aut}(T)$ acting doubly
transitively on $\partial T$ carries a natural compact Hausdorff topology
inherited from Chabauty. Some of our considerations are valid in the context of
automorphism groups of locally finite connected graphs. Applications to
Weyl-transitive automorphism groups of buildings are also presented.
