{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/saturated-fully-leafed-tree-like-polyforms","title":"Saturated Fully Leafed Tree-Like Polyforms and Polycubes","arxiv_id":"1803.09181","date":"2018-03-24","proceeding":null,"authors":["Blondin Massé Alexandre","de Carufel Julien","Goupil Alain"],"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 $3k+2$.","url_abs":"http://arxiv.org/abs/1803.09181v1","url_pdf":"http://arxiv.org/pdf/1803.09181v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"saturated-fully-leafed-tree-like-polyforms","repo_url":"https://gitlab.com/ablondin/flis-tree-polycubes","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}