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A054581 Number of unlabeled 2-trees with n nodes. 29
1, 1, 1, 2, 5, 12, 39, 136, 529, 2171, 9368, 41534, 188942, 874906, 4115060, 19602156, 94419351, 459183768, 2252217207, 11130545494, 55382155396, 277255622646, 1395731021610, 7061871805974, 35896206800034, 183241761631584 (list; graph; refs; listen; history; text; internal format)



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).


Table of n, a(n) for n=1..26.

T. Fowler, I. Gessel, G. Labelle, P. 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

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

Index entries for sequences related to trees


a(1)=a(2)=a(3)=1 because: K_2, K_3 are the only 2-trees on 2 and 3 nodes and on 4 nodes, there is a also 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.

Cf. A036361.

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

Sequence in context: A050237 A050258 A051436 * A203151 A140440 A005664

Adjacent sequences:  A054578 A054579 A054580 * A054582 A054583 A054584




Vladeta Jovovic, Apr 11 2000


Additional comments from Gordon F. Royle, Dec 02 2002



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Last modified January 24 10:01 EST 2022. Contains 350534 sequences. (Running on oeis4.)