Trees and languages with periodic signature

We first show that  Lpq has a periodic signature and the period (a sequence of q integers whose sum is p) is directly derived from the Christoffel word of slope  pq. Conversely, we give a canonical way to label a tree generated by any periodic signature; its branch language then proves to be the set of representations of the integers in a rational base (determined by the period) and written with a non-canonical alphabet of digits. This language is very much of the same kind as a  Lpq since rational base numeration systems have the key property that, even though  Lpq is not regular, normalisation is realised by a finite letter-to-letter transducer.