  Let $\mathbb T_{q+1}$ denote the homogeneous tree of degree $q+1$ with the
standard graph distance $d$ and the canonical flow measure $\mu$. The metric
measure space $(\mathbb T_{q+1},d,\mu)$ is of exponential growth. Let
$\mathcal{L}$ denote the flow Laplacian, which is a probabilistic Laplacian
self-adjoint on $L^2(\mu)$. In this note, we prove some weighted
$L^1$-estimates for the heat kernel associated with $\mathcal{L}$ and its
gradient. As a consequence, we show that the first order Riesz transform
associated with the flow Laplacian on $\mathbb T_{q+1}$ is bounded on
$L^p(\mu)$, for $p \in (1,2]$ and of weak type $(1,1)$. The latter result was
proved in a previous paper by Hebisch and Steger: we give a different proof
that might pave the way to further generalizations.
