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Regular sequences generalize the extensively studied automatic sequences. Let$S$ be an abstract numeration system. When the numeration language $L$ isprefix-closed and regular, a sequence is said to be $S$-regular if the modulegenerated by its $S$-kernel is finitely generated.  In this paper, we give a new characterization of such sequences in terms ofthe underlying numeration tree $T(L)$ whose nodes are words of $L$. We maydecorate these nodes by the sequence of interest following a breadth-firstenumeration. For a prefix-closed regular language $L$, we prove that a sequenceis $S$-regular if and only if the tree $T(L)$ decorated by the sequence islinear, i.e., the decoration of a node depends linearly on the decorations of afixed number of ancestors.  Next, we introduce and study regular sequences in a rational base numerationsystem, whose numeration language is known to be highly non-regular. Wemotivate and comment our definition that a sequence is $\frac{p}{q}$-regular ifthe underlying numeration tree decorated by the sequence is linear. We give thefirst few properties of such sequences, we provide a few examples of them, andwe propose a method for guessing $\frac{p}{q}$-regularity. Then we discuss therelationship between $\frac{p}{q}$-automatic sequences and$\frac{p}{q}$-regular sequences. We finally present a graph directed linearrepresentation of a $\frac{p}{q}$-regular sequence. Our study permits us tohighlight the places where the regularity of the numeration language plays apredominant role.

Revisiting regular sequences in light of rational base numeration systems