  Let $G_{p,d}$ be the restricted wreath product $C_{p} wr C^{d}$ where $C_{p}$
is a cyclic group of order a prime $p$ and $C^{d}$ a free abelian group of
finite rank $d$. We study the existence of faithful state-closed (fsc)
representations of $G_{p,d}$ on the $1$-rooted $m$-ary tree for some finite
$m$. The group $G_{2,1}$, known as the lamplighter group, admits an fsc
representation on the binary tree. We prove that for $d \geq 2$ there are no
fsc representations of $G_{p,d}$ on the $p$ -adic tree. We characterize all fsc
representations of $G=G_{p,1}$ on the $p$-adic tree where the first level
stabilizer of the image of $G$ contains its commutator subgroup. Furthermore,
for $d \geq 2$, we construct uniformly fsc representations of $G_{p,d}$ on the
$p^{2}$ -adic tree and exhibit concretely the representation of $G_{2,2}$ on
the $4$-tree as a finite-state automaton group.
