  In this paper we investigate some properties of the harmonic Bergman spaces
$\mathcal A^p(\sigma)$ on a $q$-homogeneous tree, where $q\geq 2$, $1\leq
p<\infty$, and $\sigma$ is a finite measure on the tree with radial decreasing
density, hence nondoubling. These spaces were introduced by J.~Cohen,
F.~Colonna, M.~Picardello and D.~Singman. When $p=2$ they are reproducing
kernel Hilbert spaces and we compute explicitely their reproducing kernel. We
then study the boundedness properties of the Bergman projector on $L^p(\sigma)$
for $1<p<\infty$ and their weak type (1,1) boundedness for radially
exponentially decreasing measures on the tree. The weak type (1,1) boundedness
is a consequence of the fact that the Bergman kernel satisfies an appropriate
integral H\"ormander's condition.
