Riesz means on homogeneous trees

Let $\mathbb{T}$ be a homogeneous tree. We prove that if $f\in L^p(X)$, $1\le p\le 2$, then the Riesz means $S_R^z\left(f\right)$ converge to $f$ almost everywhere as $R\to \mathrm{\infty }$, whenever $\mathrm{Re}z>0$.