Finally, the fourth proof of Cayley's formula presented by Aigner and Ziegler is a double counting proof due to Jim Pitman. In this proof, Pitman considers the sequences of directed edges that may be added to an "n"-node empty graph to form from it a single rooted tree, and counts the number of such sequences in two different ways. By showing how to derive a sequence of this type by choosing a tree, a root for the tree, and an ordering for the edges in the tree, he shows that there are "Tnn"! possible sequences of this type. And by counting the number of ways in which a partial sequence can be extended by a single edge, he shows that there are "n""n" − 2"n"! possible sequences. Equating these two different formulas for the size of the same set of edge sequences and cancelling the common factor of "n"! leads to Cayley's formula.