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 \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 $3k+2$.