The sandpile group of a tree

Proof. If the edge (r, s) from the root to the sink is included in the spanning tree, then each of the principal branches of Tn may be assigned an oriented spanning tree independently, so there are t n−1 such spanning trees. On the other hand, if (r, s) is not included in the spanning tree, there is a directed path r → x1 → . . . → xn−1 → s in the spanning tree from the root to the sink. In this case, every principal branch except the one rooted at x1 may be assigned an oriented spanning tree independently; within the branch rooted at x1, every subbranch except the one rooted at x2 may be assigned an oriented spanning tree independently; and so on (see Figure 1). Since there are (d − 1) possible paths x1 → . . . → xn−1, we conclude that