Enumeration of solid 2-trees according to edge number and edge degree distribution

The goal of this paper is to enumerate solid 2-trees according to the number of edges (or triangles) and also according to the edge degree distribution. We first enumerate oriented solid 2-trees using the general methods of the theory of species. In order to obtain non-oriented enumeration formulas, we use quotient species, which consists in a specialization of Pólya theory.