  We consider (a) the partition functions of the anisotropic dimer model on the
rectangular (2M-1) x (2N-1) lattice with free and cylindrical boundary
conditions with a single monomer residing on the boundary and (b) the partition
function of the anisotropic spanning tree on an M x N rectangular lattice with
free boundary conditions. We express (a) and (b) in terms of a principal
partition function with twisted boundary conditions. Based on these
expressions, we derive the exact asymptotic expansions of the free energy for
both cases (a) and (b). We confirm the conformal field theory prediction for
the corner free energy of these models, and find the central charge is c = - 2.
We also show that the dimer model on the cylinder with an odd number of sites
on the perimeter exhibits the same finite-size corrections as on the plane.
