Papers › Saturated Fully Leafed Tree-Like Polyforms and Polycubes
Saturated Fully Leafed Tree-Like Polyforms and Polycubes
Blondin Massé Alexandre, de Carufel Julien, Goupil Alain
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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 \emph{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 \emph{saturated}, which is the family of fully leafed structures that maximize the ratio /. In the polyomino case, we present a bijection between the set of saturated tree-like polyominoes of size 4k+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 41k+28 and a family of polycubes, called $4$-trees, of size 3k+2.
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