  An important problem in statistical physics concerns the fascinating
connections between partition functions of lattice models studied in
equilibrium statistical mechanics on the one hand and graph theoretical
enumeration problems on the other hand. We investigate the nature of the
relationship between the number of spanning trees and the partition function of
the Ising model on the square lattice. The spanning tree generating function
$T(z)$ gives the spanning tree constant when evaluated at $z=1$, while giving
he lattice green function when differentiated. It is known that for the
infinite square lattice the partition function $Z(K)$ of the Ising model
evaluated at the critical temperature $K=K_c$ is related to $T(1)$. Here we
show that this idea in fact generalizes to all real temperatures. We prove that
$ ( Z(K) {\rm sech~} 2K ~\!)^2 = k \exp\big[ T(k) \big] $, where $k= 2
\tanh(2K) {\rm sech}(2K)$. The identical Mahler measure connects the two
seemingly disparate quantities $T(z)$ and $Z(K)$. In turn, the Mahler measure
is determined by the random walk structure function. Finally, we show that the
the above correspondence does not generalize in a straightforward manner to
non-planar lattices.
