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A027416
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Number of unlabeled (and unrooted) trees on n nodes having a centroid.
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9
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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
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OFFSET
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0,6
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COMMENTS
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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.
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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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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.)
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FORMULA
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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
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MAPLE
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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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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