A000676 Number of centered trees with n nodes. (Formerly M0831 N0316)

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

0,6

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

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.

Formula

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

Example

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

Mathematica

See link.

Crossrefs

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

Sequence in context: A259593 A129016 A099163 * A283823 A263658 A296517

Adjacent sequences: A000673 A000674 A000675 * A000677 A000678 A000679

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Status

approved