  We prove the $L^p$-boundedness, for $p \in (1,\infty)$, of the first order
Riesz transform associated to the flow Laplacian on a homogeneous tree with the
canonical flow measure. This result was previously proved to hold for $p \in
(1,2]$ by Hebisch and Steger, but their approach does not extend to $p>2$ as we
make clear by proving a negative endpoint result for $p = \infty$ for such
operator. We also consider a class of ``horizontal Riesz transforms''
corresponding to differentiation along horocycles, which inherit all the
boundedness properties of the Riesz transform associated to the flow Laplacian,
but for which we are also able to prove a weak type $(1,1)$ bound for the
adjoint operators, in the spirit of the work by Gaudry and Sj\"ogren in the
continuous setting. The homogeneous tree with the canonical flow measure is a
model case of a measure-metric space which is nondoubling, of exponential
growth, does not satisfy the Cheeger isoperimetric inequality, and where the
Laplacian does not have spectral gap.
