  The Jones polynomial can be expressed in terms of spanning trees of the graph
obtained by checkerboard coloring a knot diagram. We show there exists a
complex generated by these spanning trees whose homology is the reduced
Khovanov homology. The spanning trees provide a filtration on the reduced
Khovanov complex and a spectral sequence that converges to its homology. For
alternating links, all differentials on the spanning tree complex are zero and
the reduced Khovanov homology is determined by the Jones polynomial and
signature. We prove some analogous theorems for (unreduced) Khovanov homology.
