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A102911
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Number of unlabeled (and unrooted) trees on 2n nodes with a bicentroid.
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8
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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
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OFFSET
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0,4
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COMMENTS
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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.
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REFERENCES
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F. Harary, Graph Theory, Addison-Wesley, Reading, MA, 1994; pp. 35, 36.
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LINKS
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FORMULA
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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).
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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