Some Exact Results for Spanning Trees on Lattices

For $n$-vertex, $d$-dimensional lattices $\mathrm{\Lambda }$ with $d\ge 2$, the number of spanning trees $N_{ST}(\mathrm{\Lambda })$ grows asymptotically as $\exp (n{z}_{\mathrm{\Lambda }})$ 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.