Enumeration of Tree-like Maps with Arbitrary Number of Vertices

This paper provides the generating series for the embedding of tree-like graphs of arbitrary number of vertices, accourding to their genus. It applies and extends the techniques of Chan, where it was used to give an alternate proof of the Goulden and Slofstra formula. Furthermore, this greatly generalizes the famous Harer-Zagier formula, which computes the Euler characteristic of the moduli space of curves, and is equivalent to the computation of one vertex maps.