We are interested in the problem of expressiveness of the following operations in the category of sets: finite products and coproducts, initial algebras (induction) and final coalgebras (coinduction). These operations are the building blocks of a logical system that allows circularity in Gentzen-style proofs. Proofs in this system are seen as simple programs, while the cut-elimination process is viewed as a running automaton with a memory device. In this paper, we show that higher-order trees, those accepted by higher-order pushdown automata, are computable in this setting, by providing an explicit simulation of the automata by cut-elimination.