The structure of tiles in ${\mathbb{Z}}_{p^n}\times {\mathbb{Z}}_q$ and ${\mathbb{Z}}_{p^n}\times {\mathbb{Z}}_p$

In this paper, we provide a geometric characterization of tiles in the finite abelian groups ${\mathbb{Z}}_{p^n}\times {\mathbb{Z}}_q$ and ${\mathbb{Z}}_{p^n}\times {\mathbb{Z}}_p$ using the concept of a $p$-homogeneous tree, which provides an intuitively visualizable criterion.