Counting components of moduli space of HCMU spheres via weighted trees

HCMU surfaces are compact Riemann surfaces equipped with the Calabi extremal K\"{a}hler metric and a finite number of singularities. By using both the classical football decomposition introduced by Chen-Chen-Wu and the description of the geometric structure of HCMU surfaces by Lu-Xu, we can use weighted plane trees to characterize HCMU spheres with a single integral conical angle. Moreover, we obtain an explicit counting formula for the components of the moduli space of such HCMU spheres by enumerating some class of weighted plane trees.