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In \cite{M}, Sebastian Mizera discovered a tree expansion formula of ahomology intersection number on the configuration space $\mathcal{M}_{0,n}$.The formula originates in a study of Kawai-Lewellen-Tye relation in stringtheory. In this paper, we give an elementary proof of the formula. The basicingredients are the combinatorics of the real moduli space$\overline{\mathcal{M}}_{0,n}(\R)$ and a combinatorial identity related to theface number of the associahedron.

A tree expansion formula of a homology intersection numbers on the configuration space $\mathcal{M}_{0,n}$