Before revision: We develop and study the structure of combinatorial objects that are a special case of RNA secondary structures. These are generalizations of objects arising from interval exchange transformations in work of J. Cassaigne, S. Ferenczi and L. Q. Zamboni. We represent them as labelled edge-coloured trees. We classify and enumerate them and show that a natural subset is in bijection with a set of m-clusters.
After revision: We study a circular order on labelled, m-edge-coloured trees with k vertices, and show that the set of such trees with a fixed circular order is in bijection with the set of RNA m-diagrams of degree k, combinatorial objects which can be regarded as RNA secondary structures of a certain kind. We enumerate these sets and show that a natural subset is in bijection with a set of m-clusters.