In contrast to the tree in the photograph Essential, this tree is hardly space-filling. As in the case of essentialness, superfluity has more to do with density than with size: a submodule is superfluous if it is, in some sense, sparse in the module. Specifically, a submodule is superfluous in M if it is so "small" that it cannot generate all of M even when taken together with any other proper R-submodule. Using trees in the photographs for the concepts of both "essential" and "superfluous" emphasizes these adjectives' natures as mathematical duals.