{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:48Z","timestamp":1753893828049,"version":"3.41.2"},"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>In this note we consider ternary trees naturally embedded in the plane in a  deterministic way. The root has position zero, or in other words label zero, and  the three children of a node with position $j\\in\\mathbb{Z}$ have positions $j-1$,  $j$,  and $j+1$. We derive the generating function of embedded ternary trees where all  internal nodes have labels less than or equal to $j$, with $j\\in\\mathbb{N}$.  Furthermore,  we study the generating function of the number of ternary trees of size $n$ with a given number of internal nodes with label $j$. Moreover, we discuss generalizations of this counting problem to several labels at the same time. We also study a refinement of the depth of the external node of rank $s$, with $0\\le s\\le 2n$, by keeping track of the left, center, and right steps on the unique path from the root to the external node. The $2n+1$ external nodes of a ternary tree are ranked from the left to the right according to an inorder traversal of the tree.  Finally, we discuss generalizations of the considered enumeration problems to embedded $d$-ary trees.<\/jats:p>","DOI":"10.37236\/629","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:44:14Z","timestamp":1578714254000},"source":"Crossref","is-referenced-by-count":1,"title":["A Note on Naturally Embedded Ternary Trees"],"prefix":"10.37236","volume":"18","author":[{"given":"Markus","family":"Kuba","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2011,7,15]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p142\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p142\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:08:16Z","timestamp":1579302496000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v18i1p142"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,7,15]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/629","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2011,7,15]]},"article-number":"P142"}}