A000676 Number of centered trees with n nodes.

1, 1, 0, 1, 1, 2, 3, 7, 12, 27, 55, 127, 284, 682, 1618, 3979, 9823, 24722, 62651, 160744, 415146, 1081107, 2831730, 7462542, 19764010, 52599053, 140580206, 377244482, 1016022191, 2745783463, 7443742141, 20239038700, 55178647926, 150820588425, 413226000775

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

A tree has either a center or a bicenter and either a centroid or a bicentroid. (These terms were introduced by Jordan.)

If the number of edges in a longest path in the tree is 2m, then the middle node in the path is the unique center, otherwise the two middle nodes in the path are the unique bicenters.

On the bottom of first page 266 of article Cayley (1881) is a table of A000676 and A000677 for n = 1..13. - Michael Somos, Aug 20 2018

Examples

			G.f. = 1 + x + x^3 + x^4 + 2*x^5 + 3*x^6 + 7*x^7 + 12*x^8 + 27*x^9 + 55*x^10 + ... - _Michael Somos_, Aug 20 2018
		

References

• N. L. Biggs et al., Graph Theory 1736-1936, Oxford, 1976, p. 49.
• F. Harary, Graph Theory, Addison-Wesley, Reading, MA, 1994; pp. 35, 36.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Geoffrey Critzer, Table of n, a(n) for n = 0..200 (replacing the first version from N. J. A. Sloane)
• Jean-François Alcover, Mathematica program
• A. Cayley, On the analytical forms called trees, with application to the theory of chemical combinations, Reports British Assoc. Advance. Sci. 45 (1875), 257-305 = Math. Papers, Vol. 9, 427-460 (see p. 438).
• A. Cayley, On the analytical forms called trees, Amer. J. Math., 4 (1881), 266-268.
• C. Jordan, Sur les assemblages des lignes, J. Reine angew. Math., 70 (1869), 185-190.
• E. M. Rains and N. J. A. Sloane, On Cayley's Enumeration of Alkanes (or 4-Valent Trees), J. Integer Sequences, Vol. 2 (1999), Article 99.1.1. [This articles states incorrectly that A000676 and A000677 give the numbers of trees with respectively a centroid and bicentroid.]
• Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 [but beware errors] (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
• Peter Steinbach, Field Guide to Simple Graphs, Volume 3, Part 12 [but beware errors] (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
• Eric Weisstein's World of Mathematics, Centered Tree.
• Index entries for sequences related to trees.

Crossrefs

Cf. A102911 (trees with a bicentroid), A027416 (trees with a centroid), A000677 (trees with a bicenter), A000055 (trees), A000081 (rooted trees).

Programs

Mathematica

(* See link. *)

Formula

a(n) + A000677(n) = A000055(n).