The Specification of 2-trees

Abstract

We derive new functional equations at a species level for certain classes of 2-trees, including a dissymmetry theorem. From these equations we deduce various series expansions for these structures. We obtain formulas for unlabeled 2-trees which are more explicit than previously known results. Moreover, the asymptotic behavior of unlabeled 2-trees is established.  2002 Elsevier Science (USA) Nous présentons de nouvelleséquations fonctionnelles pour certaines classes de 2-arbres, incluant un théorème de dissymétrie. Nous en déduisons diverses séries génératrices associéesà ces espèces. Nous obtenons ainsi des formulesénumératives pour les 2-arbres non-étiquetés qui sont plus explicites que les résultats connus jusquà présent. De plus le comportement asymptotique de ces structures estétabli.

Prime graphs and exponential composition of species

“…It appeared in [5] and [3] as an important tool in combinatorial enumeration. Suppose that a group A acts naturally (see [2, p. 393]) on a species F .…”

Graph limits of random graphs from a subset of connected k‐trees

“…The k-trees are the maximal graphs with treewidth ≤ k, in the sense that adding another edge would increase the treewidth. The number of k-trees has been counted in various ways; see [11,18,29,30,32,34,40,41,56]. As usual a graph on n vertices is called labelled if the integers from {1, 2, … , n} are assigned to its vertices (one-to-one).…”

Scaling limit of random k-trees

“…ktrees are also very interesting from a combinatorial point of view. For example, the enumeration problem for ktrees has been studied in various ways; see [7,37,24,14,31,32,25,26,27]. The number of labelled k-trees has been determined by Beineke and Pippert [7], Moon [37], Foata [24], Darrasse and Soria [14]; as usual a ktree on n vertices is called labelled if the integers from {1, 2, .…”