Fully Leafed Tree-Like Polyominoes and Polycubes

We present and prove recursive formulas giving the maximal number of leaves in tree-like polyominoes and polycubes of size n. We call these tree-like polyforms fully leafed. The proof relies on a combinatorial algorithm that enumerates rooted directed trees that we call abundant. We also show how to produce a family of fully leafed tree-like polyominoes and a family of fully leafed tree-like polycubes for each possible size, thus gaining insight into their geometric characteristics.