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

中文翻译：

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自动序列：从有理数到树

自动序列的$ n $项是确定性有限自动机的输出，该有限自动机在适当的计算系统中馈入$ n $的表示形式。在本文中，我们不考虑基于具有常规编号语言的编号系统构建的自动序列，而是考虑基于与具有周期性标签签名的树相关的语言（尤其是合理的基本编号系统）构建的自动序列。我们获得了这些序列的两个主要特征。第一个与$ r $块替换有关，其中$ r $态射定期应用。特别是，我们提供了此类非形态序列的示例。第二个特征涉及与计算系统关联并由序列项修饰的树的因子或有限高度的子树。