Some Exact Results for Spanning Trees on Lattices

For $n$-vertex, $d$-dimensional lattices $\Lambda$ with $d \ge 2$, the number of spanning trees $N_{ST}(\Lambda)$ grows asymptotically as $\exp(n z_\Lambda)$ in the thermodynamic limit. We present an exact closed-form result for the asymptotic growth constant $z_{bcc(d)}$ for spanning trees on the $d$-dimensional body-centered cubic lattice. We also give an exact integral expression for $z_{fcc}$ on the face-centered cubic lattice and an exact closed-form expression for $z_{488}$ on the $4 \cdot 8 \cdot 8$ lattice.