A027416 Number of unlabeled (and unrooted) trees on n nodes having a centroid.

1, 1, 0, 1, 1, 3, 3, 11, 13, 47, 61, 235, 341, 1301, 1983, 7741, 12650, 48629, 82826, 317955, 564225, 2144505, 3926353, 14828074, 27940136, 104636890, 201837109, 751065460, 1479817181, 5469566585, 10975442036, 40330829030, 82270184950

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Also, number of rooted unlabeled trees on n nodes not having a primary branch.

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 other hand, define the weight of a node P to be the greatest number of nodes in any subtree connected to P. Then either there is a unique node of minimal weight, the centroid of the tree, or there is a unique pair of minimal weight nodes, the bicentroids.

Let T be a tree with root node R. If R and the edges incident with it are deleted, the resulting rooted trees are called branches. A primary branch (there can be at most one) has i nodes where n/2 <= i <= n-1.

References

• F. Harary, Graph Theory, Addison-Wesley, Reading, MA, 1994; pp. 35, 36.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..200
• 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.
• A. Meir and J. W. Moon, On the branch-sizes of rooted unlabeled trees, in "Graph Theory and Its Applications", Annals New York Acad. Sci., Vol. 576, 1989, pp. 399-407.
• 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 (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 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
• Index entries for sequences related to rooted trees

Crossrefs

Cf. A102911 (trees with a bicentroid), A027415 (trees with a primary branch), A000676 (trees with a center), A000677 (trees with a bicenter), A000055 (trees), A000081 (rooted trees).

Programs

Maple

N := 50: Y := [ 1,1 ]: for n from 3 to N do x*mul( (1-x^i)^(-Y[ i ]), i=1..n-1); series(%,x,n+1); b := coeff(%,x,n); Y := [ op(Y),b ]; od: P:=n->sum(Y[n-i]*Y[i],i=1..floor(n/2)): seq(Y[n]-P(n),n=1..35); # Emeric Deutsch, Nov 21 2004

Formula

a(n) = A000055(n) - A102911(n/2) if n is even, else a(n) = A000055(n).

a(n) = A000081(n) - A027415(n). - Emeric Deutsch, Nov 21 2004

a(n) = [x^n] 1 + x/Product_{i=1..ceiling(n/2)-1} (1-x^i)^A000081(i). See Cayley link above. - Geoffrey Critzer, Jul 30 2022

Extensions

More terms from Emeric Deutsch, Nov 21 2004

Entry revised (with new definition) by N. J. A. Sloane, Feb 26 2007