  In this note we prove the classical decomposition theorems for certain
subgroups of the automorphism group of the Bruhat-Tits tree associated to
$\mathrm{PGL}_{2}(F)$, where $F$ is a local field. The results can be used to
understand the representation theory of these subgroups. Moreover, we can also
use the results to obtain similar decompositions for \textit{any} closed
transitive subgroup of the automorphism group of \textit{any} Bruhat-Tits tree.
