  For a graph G, the generating function of rooted forests, counted by the
number of connected components, can be expressed in terms of the eigenvalues of
the graph Laplacian. We generalize this result from graphs to cell complexes of
arbitrary dimension. This requires generalizing the notion of rooted forest to
higher dimension. We also introduce orientations of higher dimensional rooted
trees and forests. These orientations are discrete vector fields which lead to
open questions concerning expressing homological quantities combinatorially.
