Trees · Superchallenge problem

The $3 \times 3$ grid graph has $9$ vertices arranged in a $3 \times 3$ square: vertices $1$ through $9$ where vertex $i$ is at row $\lceil i/3 \rceil$ and column $(i-1 \bmod 3) + 1$. Edges connect adjacent vertices (horizontally and vertically). How many spanning trees does this graph have?