  We use discrete Morse theory to give a new proof of the Degree Theorem in
Auter space A_n. There is a filtration of A_n into subspaces A_{n,k} using the
degree of a graph, and the Degree Theorem says that each A_{n,k} is
(k-1)-connected. This result is useful, for example to calculate stability
bounds for the homology of Aut(F_n). The standard proof of the Degree Theorem
is global in nature. Here we give a proof that only uses local considerations,
and lends itself more readily to generalization.
