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

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

Author

Geoffrey Critzer, Aug 04 2022

Keywords

• nonn

Comments

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.

Links

• 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

Crossrefs

Cf. A000676.

Programs

Mathematica

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]