  Given a locally finite leafless tree $ T $, various algebraic groups over
local fields might appear as closed subgroups of $ \text{Aut} (T)$. We show
that the set of closed cocompact subgroups of $ \text{Aut} (T)$ that are
isomorphic to a quasi-split simple algebraic group is a closed subset of the
Chabauty space of $ \text{Aut} (T)$. This is done via a study of the integral
Bruhat$-$Tits model of $ \text{SL}_2 $ and $ \text{SU}_3^{L/K} $, that we carry
on over arbitrary local fields, without any restriction on the (residue)
characteristic. In particular, we show that in residue characteristic $ 2 $,
the Tits index of simple algebraic subgroups of $ \text{Aut} (T)$ is not always
preserved under Chabauty limits.
