  We generalize the definition and enumeration of spanning trees from the
setting of graphs to that of arbitrary-dimensional simplicial complexes
$\Delta$, extending an idea due to G. Kalai. We prove a simplicial version of
the Matrix-Tree Theorem that counts simplicial spanning trees, weighted by the
squares of the orders of their top-dimensional integral homology groups, in
terms of the Laplacian matrix of $\Delta$. As in the graphic case, one can
obtain a more finely weighted generating function for simplicial spanning trees
by assigning an indeterminate to each vertex of $\Delta$ and replacing the
entries of the Laplacian with Laurent monomials. When $\Delta$ is a shifted
complex, we give a combinatorial interpretation of the eigenvalues of its
weighted Laplacian and prove that they determine its set of faces uniquely,
generalizing known results about threshold graphs and unweighted Laplacian
eigenvalues of shifted complexes.
