Super-state automata and rational trees

Abstract

We introduce the notion of super-state automata constructed from other automata. This construction is used to solve an open question about enumerative sequences in rational trees. We prove that any N-rational sequence s = (sn)n≥0 of nonnegative integers satisfying the Kraft inequality Σn≥0 snk-n ≤1 is the enumerative sequence of leaves by height of a k-ary rational tree. This result had been conjectured and was known only in the case of strict inequality. We also give a new proof of a result about enumerative sequences of nodes in k-ary rational trees.