0,6

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.

A. Cayley, On the analytical forms called trees, Amer. J. Math., 4 (1881), 266-268.

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

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.

N. J. A. Sloane, Table of n, a(n) for n = 0..200

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.]

Index entries for sequences related to rooted trees

a(n)=A000055(n) - A102911(n).

a(n)=A000081(n) - A027415(n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 21 2004

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); (Deutsch)

Cf. A102911 (trees with a bicentroid), A027415 (trees without a primary branch), A000676 (trees with a center), A000677 (trees with a bicenter), A000055 (trees), A000081 (rooted trees).

Sequence in context: A136123 A045495 A045494 this_sequence A163932 A007022 A011950

Adjacent sequences: A027413 A027414 A027415 this_sequence A027417 A027418 A027419

nonn

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 21 2004

Entry revised (with new definition) by N. J. A. Sloane (njas(AT)research.att.com), Feb 26 2007