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