  We study Newton polytopes for cluster variables in cluster algebras
$\mathcal{A}(\Sigma)$ of types A and D. A famous property of cluster algebras
is the Laurent phenomenon: each cluster variable can be written as a Laurent
polynomial in the cluster variables of the initial seed $\Sigma$. The cluster
variable Newton polytopes are the Newton polytopes of these Laurent
polynomials. We show that if $\Sigma$ has principal coefficients or boundary
frozen variables, then all cluster variable Newton polytopes are saturated. We
also characterize when these Newton polytopes are \emph{empty}; that is, when
they have no non-vertex lattice points.
