Counting Weighted Bi-Colored Plane Trees and Their Geometric Applications

Abstract

This work solves the enumeration problem for weighted bi-colored plane trees with prescribed numbers of black and white vertices, together with prescribed total edge weights at each vertex. For the general case, we provide a unified algorithmic counting method. We then apply this result to two geometric problems. First, we compute the strong Hurwitz number for a special class of branch datum between Riemann spheres with three branched points. Second, we study the moduli space for a special class of extremal Kähler metrics on Riemann sphere (HCMU spheres), with a single conical singularity. We determine the number of its connected components with respect to the Gromov-Hausdorff topology.