  We present the detailed calculations of the asymptotics of two-site
correlation functions for height variables in the two-dimensional Abelian
sandpile model. By using combinatorial methods for the enumeration of spanning
trees, we extend the well-known result for the correlation $\sigma_{1,1} \simeq
1/r^4$ of minimal heights $h_1=h_2=1$ to $\sigma_{1,h} = P_{1,h}-P_1P_h$ for
height values $h=2,3,4$. These results confirm the dominant logarithmic
behaviour $\sigma_{1,h} \simeq (c_h\log r + d_h)/r^4 + {\cal O}(r^{-5})$ for
large $r$, predicted by logarithmic conformal field theory based on field
identifications obtained previously. We obtain, from our lattice calculations,
the explicit values for the coefficients $c_h$ and $d_h$ (the latter are new).
