{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T20:11:17Z","timestamp":1774555877535,"version":"3.50.1"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Given a tree $T$ and a subtree $S$ of $T,$ one can define the local mean at $S,$ $\\mu_{T}\\left(S\\right),$ to be the average order of the subtrees of $T$ containing $S.$ In 1983, Jamison showed that $\\mu_{T}\\left(S\\right)&lt;\\mu_{T}(S^{'})$ if $S\\subset S^{'}$ as subtrees of $T.$ Therefore, it is natural to ask the following question. Among all the $k$-subtrees (subtrees of order $k$), which one achieves the maximal\/minimal local mean and what properties does it have? We call such $k$-subtrees $k$-maximal\/$k$-minimal. Wagner and H. Wang showed in 2016 that a 1-maximal subtree has degree 1 or 2. In this paper, we show that if $T$ is not a path, a 1-minimal subtree of $T$ has degree at least 3. For $k\\geq2,$ we show that a $k$-maximal subtree has at most one leaf whose degree in $T$ is greater than 2, and that such a leaf can only occur when all other leaves in $S$ are also leaves in $T.$ Parallel results hold for $k$-minimal subtrees. Roughly speaking, the leaves of a $k$-maximal subtree tend to have degree 1 or 2 in $T,$ while the leaves of a $k$-minimal subtree tend to have degree at least 3 in $T.$\r\nIn the second part, this paper introduces the local density as a normalization of local means, for the sake of comparing subtrees of different orders. We show that the local density at subtree $S$ is lower-bounded by $1\/2$ with equality if and only if $S$ contains all the vertices of degree at least 3 in $T.$ On the other hand, local density can be arbitrarily close to 1.<\/jats:p>","DOI":"10.37236\/13813","type":"journal-article","created":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T19:48:32Z","timestamp":1774554512000},"source":"Crossref","is-referenced-by-count":0,"title":["Extrema of Local Mean and Local Density in a Tree"],"prefix":"10.37236","volume":"33","author":[{"given":"Ruoyu","family":"Wang","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2026,3,27]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i1p60\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i1p60\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T19:48:33Z","timestamp":1774554513000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v33i1p60"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,27]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1,9]]}},"URL":"https:\/\/doi.org\/10.37236\/13813","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,27]]},"article-number":"P1.60"}}