  We classify $n$-hereditary monomial algebras in three natural contexts:
First, we give a classification of the $n$-hereditary truncated path algebras.
We show that they are exactly the $n$-representation-finite Nakayama algebras
classified by Vaso. Next, we classify partially the $n$-hereditary quadratic
monomial algebras. In the case $n=2$, we prove that there are only two
examples, provided that the preprojective algebra is a planar quiver with
potential. The first one is a Nakayama algebra and the second one is obtained
by mutating $\mathbb A_3\otimes_k \mathbb A_3$, where $\mathbb A_3$ is the
Dynkin quiver of type $A$ with bipartite orientation. In the case $n\geq 3$, we
show that the only $n$-representation finite algebras are the
$n$-representation-finite Nakayama algebras with quadratic relations.
