Saturated fully leafed tree-like polyforms and polycubes

Abstract We present recursive formulas giving the maximal number of leaves in tree-like polyforms living in two-dimensional regular lattices and in tree-like polycubes in the three-dimensional cubic lattice. We call these tree-like polyforms and polycubes fully leafed . The proof relies on a combinatorial algorithm that enumerates rooted directed trees that we call abundant. In the last part, we concentrate on the particular case of polyforms and polycubes, that we call saturated , which is the family of fully leafed structures that maximize the ratio (number of leaves) / (number of cells) . In the polyomino case, we present a bijection between the set of saturated tree-like polyominoes of size 4 k + 1 and the set of tree-like polyominoes of size k . We exhibit a similar bijection between the set of saturated tree-like polycubes of size 41 k + 28 and a family of polycubes, called 4-trees, of size 3 k + 2 .