A000676 -id:A000676 - cp's OEIS frontend

A283824 Number of unlabeled (and unrooted) trees on n nodes having a node that is both a center and a centroid.

1, 1, 0, 1, 1, 2, 2, 6, 7, 20, 27, 83, 126

Offset: 0

N. J. A. Sloane, Mar 19 2017

In view of the errors in the second line of the table (compare A000676 and A283823), the entries a(11)=83 and a(12)=126 should be checked carefully.

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.)

Cf. A000676, A283823, A027416.

A356375 Number of unlabeled centered trees with n nodes that have exactly one diametral path (up to direction of traversal).

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 2, 5, 9, 21, 44, 107, 247, 607, 1465, 3649, 9087, 23059, 58831, 151832, 394074, 1030492, 2708343, 7157735, 19002282, 50676945, 135691504, 364725995, 983775878, 2662271414, 7226368722, 19670528467, 53685042694, 146879757368, 402786655780, 1106968400532

Offset: 0

Geoffrey Critzer, Aug 04 2022

nonn

A diametral path in a tree is a path of maximum length. A diametral path in a centered tree is necessarily of even length. Its endpoints are leaves and its middle point is the center of the tree. A centered tree with exactly one diametral path of length 2m can be decomposed into a rooted tree of height at most m-1 along with exactly 2 rooted trees of height exactly m-1. It appears that almost all centered trees (A000676) have exactly one diametral path.

J. Riordan, The enumeration of trees by height and diameter, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)
Index entries for sequences related to trees

Cf. A000676.

nn = 35; S[0, x_] := x; S[k_, x_] := Total[Nest[CoefficientList[Series[Product[1/(1 - x^i)^#[[i]], {i, 1, Length[#]}], {x, 0, nn}], x] &, {1}, k] Table[x^i, {i, 1, nn + 1}]]; R[0, x] := x;R[k_, x_] := S[k, x] - S[k - 1, x]; ReplacePart[ Sum[PadRight[
   CoefficientList[Series[S[m, x] (R[m, x]^2 + (R[m, x] /. x -> x^2))/2, {x, 0, nn}],x], nn + 1], {m, 0, nn/2}], 2 -> 1]

cp's OEIS Frontend

A283824 Number of unlabeled (and unrooted) trees on n nodes having a node that is both a center and a centroid.

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