

A054581


Number of unlabeled 2trees 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)



OFFSET

1,4


COMMENTS

A 2tree is recursively defined as follows: K_2 is a 2tree and any 2tree on n+1 vertices is obtained by joining a vertex to a 2clique in a 2tree on n vertices. Care is needed with the term 2tree (and ktree 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 3gonal 2trees with n 3gons.


REFERENCES

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 327328.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 76, t(x), (3.5.19).


LINKS

Table of n, a(n) for n=1..26.
T. Fowler, I. Gessel, G. Labelle, P. Leroux, The specification of 2trees, Adv. Appl. Math. 28 (2) (2002) 145168, Table 1.
Andrew GainerDewar, GammaSpecies and the Enumeration of kTrees, 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 kgonal 2trees, arXiv:math/0312424 [math.CO], 2003.
Index entries for sequences related to trees


EXAMPLE

a(1)=a(2)=a(3)=1 because: K_2, K_3 are the only 2trees 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 nonshared edges.


CROSSREFS

Column k=3 of A340811.
Cf. A036361.
Cf. A000272 (labeled trees), A036361 (labeled 2trees), A036362 (labeled 3trees), A036506 (labeled 4trees), A000055 (unlabeled trees).
Sequence in context: A050237 A050258 A051436 * A203151 A140440 A005664
Adjacent sequences: A054578 A054579 A054580 * A054582 A054583 A054584


KEYWORD

nonn,nice


AUTHOR

Vladeta Jovovic, Apr 11 2000


EXTENSIONS

Additional comments from Gordon F. Royle, Dec 02 2002


STATUS

approved



