A combinatorial proof of a formula of Biane and Chapuy.

Let G be a simple strongly connected weighted directed graph. Let G denote the spanning tree graph of G. That is, the vertices of G consist of the directed rooted spanning trees on G, and the edges of G consist of pairs of trees (t(i),t(j)) such that t(j) can be obtained from t(i) by adding the edge from the root of t(i) to the root of t(j) and deleting the outgoing edge from t(j). A formula for the ratio of the sum of the weights of the directed rooted spanning trees on G to the sum of the weights of the directed rooted spanning trees on G was recently given by Biane and Chapuy. Our main contribution is an alternative proof of this formula, which is both simple and combinatorial.