  This work deals with the characterization of eigenfunctions of the Laplacian
$\mathcal{L}$ on a homogeneous tree $\mathcal{X}$, which satisfy certain growth
conditions. More precisely, we shall prove that the Poisson transform on
$\mathcal{X}$ provides an one-to-one correspondence between the subspace of all
Hardy-type eigenfunctions of $\mathcal{L}$ on $\mathcal{X}$ and the Lebesgue
spaces (possibly the set of all complex measures) on the boundary of
$\mathcal{X}$.
