  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.
