{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:41Z","timestamp":1753893821776,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Consider a rooted binary tree with $n$ nodes. Assign with the root the abscissa 0, and with the left (resp. right) child of a node of abscissa $i$ the abscissa $i-1$ (resp. $i+1$). We prove that the number of binary trees of size $n$ having exactly $n_i$ nodes at abscissa $i$, for $l \\leq i \\leq r$ (with $n = \\sum_i n_i$), is $$ \\frac{n_0}{n_l n_r} {{n_{-1}+n_1} \\choose {n_0-1}} \\prod_{l\\le i\\le r \\atop i\\not = 0}{{n_{i-1}+n_{i+1}-1} \\choose {n_i-1}}, $$ with $n_{l-1}=n_{r+1}=0$. The sequence $(n_l, \\dots, n_{-1};n_0, \\dots n_r)$ is called the vertical profile of the tree. The vertical profile of a uniform random tree of size $n$ is known to converge, in a certain sense and after normalization, to a random mesure called the integrated superbrownian excursion, which motivates our interest in the profile. We prove similar looking formulas for other families of trees whose nodes are embedded in $Z$. We also refine these formulas by taking into account the number of nodes at abscissa j whose parent lies at abscissa $i$, and\/or the number of vertices at abscissa i having a prescribed number of children at abscissa $j$, for all $i$ and $j$. Our proofs are bijective.<\/jats:p>","DOI":"10.37236\/2150","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:23:32Z","timestamp":1578713012000},"source":"Crossref","is-referenced-by-count":0,"title":["The Vertical Profile of Embedded Trees"],"prefix":"10.37236","volume":"19","author":[{"given":"Mireille","family":"Bousquet-M\u00e9lou","sequence":"first","affiliation":[]},{"given":"Guillaume","family":"Chapuy","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2012,10,11]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i3p46\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i3p46\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T22:29:08Z","timestamp":1579300148000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v19i3p46"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,10,11]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2012,7,12]]}},"URL":"https:\/\/doi.org\/10.37236\/2150","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2012,10,11]]},"article-number":"P46"}}