A102911 Number of unlabeled (and unrooted) trees on 2n nodes with a bicentroid.

0, 1, 1, 3, 10, 45, 210, 1176, 6670, 41041, 258840, 1697403, 11359761, 77956341, 543625851, 3855429766, 27702225271, 201515674128, 1481195012220, 10991843660826, 82256068767106, 620288742329028, 4709854127998971, 35987845277616940, 276563426284762620

Offset: 0

Author

N. J. A. Sloane and David Applegate, Feb 26 2007

Keywords

• nonn

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 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.

A 2n-node tree with a bicentroid consists of two n-node rooted trees with the roots joined by an edge.

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.
• 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.]

Crossrefs

Cf. A027416 (trees with a centroid), A000676 (trees with a center), A000677 (trees with a bicenter), A000055 (trees), A000081 (rooted trees).

Formula

a(n) = r(n)*(r(n)+1)/2 where r(n) = A000081(n) is the number of rooted trees on n nodes.

Let f(n) = a(n/2) if n is even, = 0 otherwise. Then f(n) + A027416(n) = A000055(n).