There are n cells, drawn on a square grid, pointwise connected; polyominoes may be rotated by 90 degrees and turned over.
Comments from Joseph Myers, Oct 27 2003. "There is a figure for n=5 (the first term this differs from A030222) on the fourth link. I think the following explains this sequence, but someone should do the computations to verify it (and probably compute counts for "fixed" shapes - orientation matters - and one-sided shapes - at the same time and add those sequences if not present).
"Consider a polyplet (A030222) as made up of n components which are polyominoes, those polyominoes being joined to each other only at corners. Then sever all but n-1 of the diagonal links in such a way that a spanning tree remains. The present sequence counts such spanning trees (where different orientations of the same spanning tree do not count as distinct; note that a single symmetrical polyplet can produce multiple identical spanning trees of lesser symmetry in different orientations, which count as the same).
"Similarly, A056841 appears to count spanning trees of polyominoes (ordinary polyominoes, A000105), where the edges shared by two squares are the edges of the graph for the purposes of forming the spanning tree and A056787 _may_ count spanning trees of polyplets where the graph has edges joining every pair of squares that share an edge or vertex (this definitely needs computations, but it does match the first three terms)."