Automatic sequences: from rational bases to trees

The $n$th term of an automatic sequence is the output a deterministic finite automaton fed with representation $n$ in suitable numeration system. In this paper, instead considering sequences built on system regular language, we consider those languages associated trees having periodic labeled signatures and, particular, rational base systems. We obtain two main characterizations these sequences. first one concerned $r$-block substitutions where $r$ morphisms are applied periodically. provide examples such that not morphic. second characterization involves factors, or subtrees height, tree and decorated by terms sequence.