  If $G$ is a group acting on a tree $X$, and ${\mathcal S}$ is a
$G$-equivariant sheaf of vector spaces on $X$, then its compactly-supported
cohomology is a representation of $G$. Under a finiteness hypothesis, we prove
that if $H_c^0(X, {\mathcal S})$ is an irreducible representation of $G$, then
$H_c^0(X, {\mathcal S})$ arises by induction from a vertex or edge stabilizing
subgroup.
  If $G$ is a reductive group over a nonarchimedean local field $F$, then
Schneider and Stuhler realize every irreducible supercuspidal representation of
$G(F)$ in the degree-zero cohomology of a $G(F)$-equivariant sheaf on its
reduced Bruhat-Tits building $X$. When the derived subgroup of $G$ has relative
rank one, $X$ is a tree. An immediate consequence is that every such
irreducible supercuspidal representation arises by induction from a
compact-mod-center open subgroup.
