The logarithm of the number of Eulerian orientations, normalised by the number of vertices, is known as the residual entropy in studies of ice-type models on graphs. The spanning tree entropy depends similarly on the number of spanning trees. We demonstrate and investigate a remarkably strong, though non-deterministic, correlation between these two entropies. This leads us to propose a new heuristic estimate for the residual entropy of regular graphs that performs much better than previous heuristics. We also study the expansion properties and residual entropy of random graphs with given degrees.