A bijection for triangulations of a polygon with interior points and multiple edges

Loopless triangulations of a polygon with k vertices in k+2n triangles (with interior points and possibly multiple edges) were enumerated by Mullin in 1965, using generating functions and calculations with the quadratic method. In this article we propose a simple bijective interpretation of Mullin's formula. The argument rests on the method of conjugacy classes of trees, a variation of the cycle lemma designed for planar maps. In the much easier case of loopless triangulations of the sphere (k = 3), we recover and prove correct an unpublished construction of the second author.