Mixed cobinary trees

Abstract

We develop basic cluster theory from an elementary point of view using a variation of binary trees which we call mixed cobinary trees. We show that the number of isomorphism classes of such trees is given by the Catalan number Cn where n is the number of internal nodes. We also consider the corresponding quiver Q_{\epsilon} of type An-1. As a special case of more general known results about the relation between c-vectors, representations of quivers and their semi-invariants, we explain the bijection between mixed cobinary trees and the vertices of the generalized associahedron corresponding to the quiver Q_{\epsilon}.Comment: 17 pages, 8 figures, v2: definition of cluster is fixed, v3: references added, Corollary 1 and referenced to it are reworde