Joint distribution of the number of vertices with given different outdegrees in Galton-Watson forest

A Galton-Watson forest consisting of N roots (or trees) and n non-root vertices is considered. We study limit distributions of the number of vertices of a given outdegree in such a forest. In this work joint distribution of the number of vertices with the given different outdegrees r1 and r2 is considered. A local limit theorem for this characteristic is proved.