We study generalized inner functions on a large family of Reproducing Kernel Hilbert Spaces. We show that the only inner functions that are entire are the normalized monomials.

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, $${\mathbb {Z}}_{p}$$ Z p and $${\mathbb {Z}}_{q}$$ Z q , endowed with their corresponding standard non-archimedean metric topologies.