{"paper":{"title":"Relation between the number of leaves of a tree and its diameter","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Pu Qiao, Xingzhi Zhan","submitted_at":"2019-04-27T11:52:14Z","abstract_excerpt":"Let $L(n,d)$ denote the minimum possible number of leaves in a tree of order $n$ and diameter $d.$ In 1975 Lesniak gave the lower bound\n  $B(n,d)=\\lceil 2(n-1)/d\\rceil$ for $L(n,d).$ When $d$ is even, $B(n,d)=L(n,d).$ But when $d$ is odd, $B(n,d)$ is smaller than $L(n,d)$ in general.\n  For example, $B(21,3)=14$ while $L(21,3)=19.$ We prove that for $d\\ge 2,$\n  $ L(n,d)=\\left\\lceil \\frac{2(n-1)}{d}\\right\\rceil$ if $d$ is even and $L(n,d)=\\left\\lceil \\frac{2(n-2)}{d-1}\\right\\rceil$ if $d$ is odd.\n  The converse problem is also considered. Let $D(n,f)$ be the minimum possible diameter of a tree o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.12150","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":""},"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"}