{"paper":{"title":"Exact Enumeration of Phylogenetic Networks: The Tree-Child, Reticulation-Visible and Orchard Hierarchy","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["q-bio.PE"],"primary_cat":"math.CO","authors_text":"Josep Batle","submitted_at":"2026-06-23T09:02:23Z","abstract_excerpt":"We develop a unified framework for the exact enumeration and asymptotic analysis of the three most studied classes of phylogenetic networks: tree-child (TC), reticulation-visible (RV) and orchard networks, whose cardinalities satisfy the strict ordering $|\\mathrm{TC}_{\\ell,k}|<|\\mathrm{RV}_{\\ell,k}|<|\\mathrm{Orch}_{\\ell,k}|$ for reticulation number $k\\geq2$ (with $\\mathrm{TC}\\subsetneq\\mathrm{RV}$ and $\\mathrm{TC}\\subsetneq\\mathrm{Orch}$, while $\\mathrm{RV}$ and $\\mathrm{Orch}$ are incomparable as sets). Using the Chang--Fuchs structural theorem, we derive a two-level master functional equatio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.24325","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.24325/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}