Rhythmic generation of infinite trees and languages

This work builds on the notion of breadth-first signature of infinite trees and (prefix-closed) languages introduced by the authors in a previous work. We focus here on periodic signatures, a case coming from the study of rational base numeration systems; the language of integer representations in base~$\frac{p}{q}$ has a purely periodic signature whose period is derived from the Christoffel word of slope~$\frac{p}{q}$. Conversely, we characterise languages whose signature are purely periodic as representations of integers in such number systems with non-canonical alphabets of digits.