  The main focus of this contribution is on the harmonic Bergman spaces
$\mathcal{B}_{\alpha}^{p}$ on the $q$-homogeneous tree $\mathfrak{X}_q$ endowed
with a family of measures $\sigma_\alpha$ that are constant on the horocycles
tangent to a fixed boundary point and turn out to be doubling with respect to
the corresponding horocyclic Gromov distance. A central role is played by the
reproducing kernel Hilbert space $\mathcal{B}_{\alpha}^{2}$ for which we find a
natural orthonormal basis and formulae for the kernel. We also consider the
atomic Hardy space and the bounded mean oscillation space. Appealing to an
adaptation of Calder\'on-Zygmund theory and to standard boundedness results for
integral operators on $L^p_\alpha$ spaces with H\"ormander-type kernels, we
determine the boundedness properties of the Bergman projection.
