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To every automorphism w of an infinite rooted regular binary tree weassociate a two variable generating function \Phi_w that encodes information onthe orbit structure of w. We prove that this is a rational function if w can bedescribed by finitely many recursion relations of a particular form. We showthat this condition is satisfied for all elements of the discrete iteratedmonodromy group \Gamma associated to a postcritically finite quadraticpolynomial over C. For such \Gamma we also prove that there are only finitelymany possibilities for the denominator of \Phi_w, and we describe a procedureto determine their lowest common denominator.

Orbit length generating functions of automorphisms of a rooted regular binary tree