1,5

A 2-tree is recursively defined as follows: K_2 is a 2-tree and any 2-tree on n+1 vertices is obtained by joining a vertex to a 2-clique in a 2-tree on n vertices. Care is needed with the term 2-tree (and k-tree in general) because it has at least two commonly used definitions.

A036361 gives the labeled version of this sequence, which has an easy formula analogous to Cayley's formula for the number of trees.

Also, number of unlabeled 3-gonal 2-trees with n 3-gons.

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 327-328.

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 76, t(x), (3.5.19).

Carlos A. Alfaro, Jesús Uriel Medrano, and Iván Téllez Téllez, The Smith normal form of distance matrices of high dimensional trees, Comput. Appl. Math. 45 (2026), 226. See also arXiv:2510.04471 [math.CO], 2025.

Barbara Betti, Viktoriia Borovik, Bella Finkel, Bernd Sturmfels, and Bailee Zacovic, Graphical Scattering Equations, arXiv:2511.07316 [math.CO], 2025. See p. 18.

Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.

Nick Early, Anaëlle Pfister, and Bernd Sturmfels, Minimal Kinematics on M_{0,n}, arXiv:2402.03065 [math.AG], 2024.

Tom Fowler, Ira Gessel, Gilbert Labelle, and Pierre Leroux, The specification of 2-trees, Adv. Appl. Math. 28 (2) (2002) 145-168, Table 1.

Andrew Gainer-Dewar, Gamma-Species and the Enumeration of k-Trees, Electronic Journal of Combinatorics, Volume 19 (2012), #P45. - From N. J. A. Sloane, Dec 15 2012

Gilbert Labelle, Cédric Lamathe, and Pierre Leroux, Labeled and unlabeled enumeration of k-gonal 2-trees, arXiv:math/0312424 [math.CO], 2003.

Eric Weisstein's World of Mathematics, k-Tree.

a(1)=0 because K_1 is not a 2-tree;

a(2)=a(3)=1 because K_2 and K_3 are the only 2-trees on those sizes.

a(4)=1 because there is a unique example obtained by joining a triangle to K_3 along an edge (thus forming K_4\e). The two graphs on 5 nodes are obtained by joining a triangle to K_4\e, either along the shared edge or along one of the non-shared edges.

Column k=3 of A340811, column k=2 of A370770.

Cf. A000272 (labeled trees), A036361 (labeled 2-trees), A036362 (labeled 3-trees), A036506 (labeled 4-trees), A000055 (unlabeled trees).

Sequence in context: A383829 A051436 A378004 * A203151 A140440 A005664

Adjacent sequences: A054578 A054579 A054580 * A054582 A054583 A054584

nonn,nice

Vladeta Jovovic, Apr 11 2000

Additional comments from Gordon Royle, Dec 02 2002

Missing initial term 0 inserted by Brendan McKay, Aug 07 2023

approved