Tilings of the infinite $p$-ary tree and Cantor homeomorphisms

We define a notion of tiling of the full infinite $p$-ary tree, establishing a series of equivalent criteria for a subtree to be a tile, each of a different nature; namely, geometric, algebraic, graph-theoretic, order-theoretic, and topological. We show how these results can be applied in a straightforward and constructive manner to define homeomorphisms between two given spaces of $p$-adic integers, ${\mathbb{Z}}_p$ and ${\mathbb{Z}}_q$, endowed with their corresponding standard non-archimedean metric topologies.