Classroom note: An easy bijective proof of the matrix-forest theorem

The Matrix-Forest Theorem the number of C>'-UUiiJL<hv.UA. obtained A J-cycle in G, for J ~ is a directed For any node i, an i-rooted tree in we mean a tree where one outgoing which has none. In other in the tree, the path between j and i in the tree directed towards tree is sometimes called an "in-directed arborescence".) If the tree reaches all the nodes m it is an i-rooted tree. For any node set I ~ by I-rooted spanning forest in G we mean a collection of i-rooted trees, one for each i E I, such that every node in V is in exactly one of