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arXiv · 1803.09181

Saturated Fully Leafed Tree-Like Polyforms and Polycubes

Blondin Massé Alexandre·de Carufel Julien·Goupil Alain
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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 \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 $\mbox{(number of leaves)}/\mbox{ (number of cells)}$. 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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Blondin Massé Alexandre, de Carufel Julien, Goupil Alain. 2018-03-24. Saturated Fully Leafed Tree-Like Polyforms and Polycubes. https://arxiv.org/abs/1803.09181

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