{"paper":{"title":"A Generalized Enumeration of Labeled Trees and Reverse Pr\\\"ufer Algorithm","license":"","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Heesung Shin, Seunghyun Seo","submitted_at":"2005-12-31T06:14:56Z","abstract_excerpt":"A {\\em leader} of a tree $T$ on $[n]$ is a vertex which has no smaller descendants in $T$. Gessel and Seo showed $$\\sum_{T \\in \\mathcal{T}_n}u^\\text{(# of leaders in $T$)} c^\\text{(degree of 1 in $T$)}=u P_{n-1}(1,u,cu),$$ which is a generalization of Cayley formula, where $\\mathcal{T}_n$ is the set of trees on $[n]$ and $$P_n(a,b,c)=c\\prod_{i=1}^{n-1}(ia+(n-i)b+c).$$ Using a variation of Pr\\\"ufer code which is called a {\\em RP-code}, we give a simple bijective proof of Gessel and Seo's formula."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0601009","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/math/0601009/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"}