Moduli spaces in genus zero and inversion of power series

Let M 0,n denote the moduli space Riemann surfaces of genus 0 with n ordered marked points. Its Deligne-Mumford compactification M 0,n is naturally partitioned into connected strata of the form indexed by the different topological types of stable curves with n marked points. The stable curves in the stratum above have s irreducible components and s − 1 nodes; thus n i = n + 2s − 2. This note provides a short proof of the following result, which shows that the universal formula for inversion of power series is encoded in the stratification of moduli space.