A000676 -id:A000676 - cp's OEIS frontend

A000055 Number of trees with n unlabeled nodes.

1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235, 551, 1301, 3159, 7741, 19320, 48629, 123867, 317955, 823065, 2144505, 5623756, 14828074, 39299897, 104636890, 279793450, 751065460, 2023443032, 5469566585, 14830871802, 40330829030, 109972410221, 300628862480, 823779631721, 2262366343746, 6226306037178

Offset: 0

N. J. A. Sloane

Also, number of unlabeled 2-gonal 2-trees with n-1 2-gons, for n>0. [Corrected by Andrei Zabolotskii, Jul 29 2025]
Main diagonal of A054924.
Left border of A157905. - Gary W. Adamson, Mar 08 2009
From Robert Munafo, Jan 24 2010: (Start)
Also counts classifications of n items that require exactly n-1 binary partitions; see Munafo link at A005646, also A171871 and A171872.
The 11 trees for n = 7 are illustrated at the Munafo web link.
Link to A171871/A171872 conjectured by Robert Munafo, then proved by Andrew Weimholt and Franklin T. Adams-Watters on Dec 29 2009. (End)
This is also "Number of tree perfect graphs on n nodes" [see Hougardy]. - N. J. A. Sloane, Dec 04 2015
For n > 0, a(n) is the number of ways to arrange n-1 unlabeled non-intersecting circles on a sphere. - Vladimir Reshetnikov, Aug 25 2016
All trees for n=1 through n=12 are depicted in Chapter 1 of the Steinbach reference. On p. 6 appear encircled two trees (with n=10) which seem inequivalent only when considered as ordered (planar) trees. Earlier instances of such possibly (in)equivalent trees could appear from n=6 on (and from n=9 on without equivalence modulo plane symmetry) but are not drawn separately there. - M. F. Hasler, Aug 29 2017

			a(1) = 1 [o]; a(2) = 1 [o-o]; a(3) = 1 [o-o-o];
a(4) = 2 [o-o-o and o-o-o-o]
            |
            o
G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 11*x^7 + 23*x^8 + ...

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 279.
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 55.
N. L. Biggs et al., Graph Theory 1736-1936, Oxford, 1976, p. 49.
A. Cayley, On the analytical forms called trees, with application to the theory of chemical combinations, Reports British Assoc. Advance. Sci. 45 (1875), 257-305 = Math. Papers, Vol. 9, 427-460 (see p. 459).
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 295-316.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 526.
F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 232.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 58 and 244.
D. E. Knuth, Fundamental Algorithms, 3d Ed. 1997, pp. 386-88.
R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 138.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. J. Mathar and Robert G. Wilson v, Table of n, a(n) for n = 0..1000
Jeremy C. Adcock, Sam Morley-Short, Joshua W. Silverstone, and Mark G. Thompson, Hard limits on the postselectability of optical graph states, arXiv:1806.03263 [quant-ph], 2018.
H. R. Afshar, E. A. Bergshoeff, and W. Merbis, Interacting spin-2 fields in three dimensions, arXiv preprint arXiv:1410.6164 [hep-th], 2014-2015.
Ruggero Bandiera and Florian Schaetz, Eulerian idempotent, pre-Lie logarithm and combinatorics of trees, arXiv:1702.08907 [math.CO], 2017. Mentions this sequence.
Madeleine Burkhart and Joel Foisy, Enumerating spherical n-links, Involve, Vol. 11 (2018), No. 2, 195-206.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
A. Cayley, On the analytical forms called trees, Amer. J. Math., 4 (1881), 266-268.
CombOS - Combinatorial Object Server, Generate graphs
Benjamin Côté, Hélène Cossette, and Etienne Marceau, Centrality and topology properties in a tree-structured Markov random field, arXiv:2410.20240 [math.ST], 2024. See p. 18.
Antoine Dailly and Tuomo Lehtilä, Reconstructing graphs with subgraph compositions, arXiv:2504.00169 [math.CO], 2025. See p. 28.
R. Ferrer-i-Cancho, Non-crossing dependencies: least effort, not grammar, arXiv preprint arXiv:1411.2645 [cs.CL], 2014.
S. R. Finch, Otter's Tree Enumeration Constants
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 481.
Andrew Gainer-Dewar, Gamma-Species and the Enumeration of k-Trees, Electronic Journal of Combinatorics, Volume 19 (2012), #P45 and arXiv:1208.5993 [math.CO], 2012.
Xiangrui Gao, Song He, and Yong Zhang, Labelled tree graphs, Feynman diagrams and disk integrals arxiv:1708.08701 [hep-th], 2017, see p. 4.
Ira M. Gessel, Good Will Hunting's Problem: Counting Homeomorphically Irreducible Trees, arXiv:2305.03157 [math.CO], 2023.
D. D. Grant, The stability index of graphs, pp. 29-52 of Combinatorial Mathematics (Proceedings 2nd Australian Conf.), Lect. Notes Math. 403, 1974. Gives first 45 terms.
Paul E. Gunnells, Generalized Catalan numbers from hypergraphs, arXiv:2102.05121 [math.CO], 2021. Mentions this sequence p. 3.
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
S. Hougardy, Classes of perfect graphs, Discr. Math. 306 (2006), 2529-2571.
House of Graphs, Trees
Andrew Jobbings, Enumerating nets, Preprint 2015.
Mithra Karamchedu and Lucas Bang, Generating the Spanning Trees of Series-Parallel Graphs up to Graph Automorphism, arXiv:2508.13480 [cs.DS], 2025. See p. 4.
Elena V. Konstantinova and Maxim V. Vidyuk, Discriminating tests of information and topological indices. Animals and trees, J. Chem. Inf. Comput. Sci. 43 (2003), 1860-1871. See Table 15, column 1 on page 1868.
G. Labelle, C. Lamathe and P. Leroux, Labeled and unlabeled enumeration of k-gonal 2-trees, arXiv:math/0312424 [math.CO], 2003.
Steve Lawford and Yll Mehmeti, Cliques and a new measure of clustering: with application to U.S. domestic airlines, arXiv:1806.05866 [cs.SI], 2018.
P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
Gang Li, Generation of Rooted Trees and Free Trees, Comp. Sci. MSc thesis paper, 1996.
R. J. Mathar, Topologically Distinct Sets of Non-intersecting Circles in the Plane, arXiv:1603.00077 [math.CO], 2016.
Richard J. Mathar, Counting Connected Graphs without Overlapping Cycles, arXiv:1808.06264 [math.CO], 2018.
Robert Munafo, Relation to Tree Graphs
R. Otter, The number of trees, Ann. of Math. (2) 49 (1948), 583-599.
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
E. M. Palmer and A. J. Schwenk, On the number of trees in a random forest, J. Combin. Theory, B 27 (1979), 109-121.
Sebastian Piec, Krzysztof Malarz and Krzysztof Kulakowski. How to count trees?, arXiv:cond-mat/0501594 [cond-mat.stat-mech], 2005; Int. J. Modern Phys., C16 (2005) 1527-1534.
N. Pippenger, Enumeration of equicolorable trees, SIAM J. Discrete Math., 14 (2001), 93-115.
J. M. Plotkin and J. W. Rosenthal, How to obtain an asymptotic expansion of a sequence from an analytic identity satisfied by its generating function, J. Austral. Math. Soc. (Series A), Vol. 56 (1994), 131-143.
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.
R. W. Robinson, Letter to N. J. A. Sloane, Jul 29 1980
Yaniv Sadeh, Haim Kaplan, and Uri Zwick, Search Trees on Trees via LP, arXiv:2501.17563 [cs.DS], 2025. See p. 12.
Sage, Common Graphs
A. J. Schwenk, Letter to N. J. A. Sloane, Aug 1972
N. J. A. Sloane, Illustration of initial terms
Peter Steinbach, Field Guide to Simple Graphs, Volume 3, Overview of the following parts: Cover, Front matter, Chapter 1: Trees, Trees (cont'd: pt.2), Trees (cont'd: pt.3), Chapter 2: Centers and Centroids, Chap.2 (cont'd), Chapter 3: Random Trees, Chapter 4: Rooted Trees, Chapter 5: Homeomorphically Irreducible Trees, Chapter 6: Tables (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Ivan Stošić and Ivan Damnjanović, An efficient algorithm for generating transmission irregular trees, arXiv:2502.15453 [cs.DM], 2025. See p. 11.
Tanay Wakhare, Eric Wityk, and Charles R. Johnson. The proportion of trees that are linear, Discrete Mathematics 343.10 (2020): 112008. Also arXiv:1901.08502 [math.CO], 2019-2020. See Tables 1 and 2 (but beware errors).
Eric Weisstein's World of Mathematics, Tree
Pascal Welke, Tamás Horváth, and Stefan Wrobel, Probabilistic and exact frequent subtree mining in graphs beyond forests, Machine Learning (2019), 1-28.
Robert Alan Wright, Bruce Richmond, Andrew Odlyzko, and Brendan D. McKay, Constant Time Generation of Free Trees, SIAM Journal of Computing, vol. 15, no. 2, pp. 540-548, (1986) [preprint].
Index entries for sequences related to trees
Index entries for "core" sequences

Cf. A000676 (centered trees), A000677 (bicentered trees), A027416 (trees with a centroid), A102011 (trees with a bicentroid), A034853 (refined by diameter), A238414 (refined by maximum vertex degree).
Cf. A000081 (rooted trees), A000272 (labeled trees), A000169 (labeled rooted trees), A212809 (radius of convergence).
Cf. A036361 (labeled 2-trees), A036362 (labeled 3-trees), A036506 (labeled 4-trees), A054581 (unlabeled 2-trees).
Cf. A157904, A157905, A005195 (Euler transform = forests), A095133 (multisets).
Column 0 of A335362 and A034799.
Related to A005646; see A171871 and A171872.
Cf. also A000088, A008406, A051491, A086308.

import Data.List (generic_index)
import Math.OEIS (getSequenceByID)
triangle x = (x * x + x) `div` 2
a000055 n = let {r = genericIndex (fromJust (getSequenceByID "A000081")); (m, nEO) = divMod n 2}
            in  r n - sum (zipWith (*) (map r [0..m]) (map r [n, n-1..]))
                + (1-nEO) * (triangle (r m + 1))
-- Walt Rorie-Baety, Jun 12 2021

N := 30; P := PowerSeriesRing(Rationals(),N+1); f := func< A | x*&*[Exp(Evaluate(A,x^k)/k) : k in [1..N]]>; G := x; for i in [1..N] do G := f(G); end for; G000081 := G; G000055 := 1 + G - G^2/2 + Evaluate(G,x^2)/2; A000055 := Eltseq(G000055); // Geoff Baileu (geoff(AT)maths.usyd.edu.au), Nov 30 2009

G000055 := series(1+G000081-G000081^2/2+subs(x=x^2,G000081)/2,x,31); A000055 := n->coeff(G000055,x,n); # where G000081 is g.f. for A000081 starting with n=1 term
with(numtheory): b:= proc(n) option remember; `if`(n<=1, n, (add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/ (n-1)) end: a:= n-> `if`(n=0, 1, b(n) -(add(b(k) *b(n-k), k=0..n) -`if`(irem(n, 2)=0, b(n/2), 0))/2):
seq(a(n), n=0..50);
# Alois P. Heinz, Aug 21 2008
# Program to create b-file b000055.txt:
A000081 := proc(n) option remember; local d, j;
if n <= 1 then n else
    add(add(d*procname(d),d=numtheory[divisors](j))*procname(n-j),j=1..n-1)/(n-1);
fi end:
A000055 := proc(nmax) local a81, n, t, a, j, i ;
a81 := [seq(A000081(i), i=0..nmax)] ; a := [] ;
for n from 0 to nmax do
    if n = 0 then
        t := 1+op(n+1, a81) ;
    else
        t := op(n+1, a81) ;
    fi;
    if type(n, even) then
        t := t-op(1+n/2, a81)^2/2 ;
        t := t+op(1+n/2, a81)/2 ;
    fi;
    for j from 0 to (n-1)/2 do
        t := t-op(j+1, a81)*op(n-j+1, a81) ;
    od:
    a := [op(a), t] ;
od:
a end:
L := A000055(1000) ;
#  R. J. Mathar, Mar 06 2009

s[n_, k_] := s[n, k] = a[n + 1 - k] + If[n < 2k, 0, s[n - k, k]]; a[1] = 1; a[n_] := a[n] = Sum[a[i] s[n-1, i] i, {i, 1, n-1}] / (n-1); Table[a[i] - Sum[a[j] a[i-j], {j, 1, i/2}] + If[OddQ[i], 0, a[i/2] (a[i/2] + 1)/2], {i, 1, 50}] (* Robert A. Russell *)
b[0] = 0; b[1] = 1; b[n_] := b[n] = Sum[d*b[d]*b[n-j], {j, 1, n-1}, {d, Divisors[j]}]/(n-1); a[0] = 1; a[n_] := b[n] - (Sum[b[k]*b[n-k], {k, 0, n}] - If[Mod[n, 2] == 0, b[n/2], 0])/2; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 09 2014, after Alois P. Heinz *)

{a(n) = local(A, A1, an, i, t); if( n<2, n>=0, an = Vec(A = A1 = 1 + O('x^n)); for(m=2, n, i=m\2; an[m] = sum(k=1, i, an[k] * an[m-k]) + (t = polcoeff( if( m%2, A *= (A1 - 'x^i)^-an[i], A), m-1))); t + if( n%2==0, binomial( -polcoeff(A, i-1), 2)))}; /* Michael Somos */

N=66;  A=vector(N+1, j, 1);
for (n=1, N, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d * A[d]) * A[n-k+1] ) );
A000081=concat([0], A);
H(t)=subst(Ser(A000081, 't), 't, t);
x='x+O('x^N);
Vec( 1 + H(x) - 1/2*( H(x)^2 - H(x^2) ) )
\\ Joerg Arndt, Jul 10 2014

# uses function from A000081
def A000055(n): return 1 if n == 0 else A000081(n)-sum(A000081(i)*A000081(n-i) for i in range(1,n//2+1)) + (0 if n % 2 else (A000081(n//2)+1)*A000081(n//2)//2) # Chai Wah Wu, Feb 03 2022

[len(list(graphs.trees(n))) for n in range(16)] # Peter Luschny, Mar 01 2020

G.f.: A(x) = 1 + T(x) - T^2(x)/2 + T(x^2)/2, where T(x) = x + x^2 + 2*x^3 + ... is the g.f. for A000081.
a(n) ~ A086308 * A051491^n * n^(-5/2). - Vaclav Kotesovec, Jan 04 2013
a(n) = A000081(n) - A217420(n+1), n > 0. - R. J. Mathar, Sep 19 2016
a(n) = A000676(n) + A000677(n). - R. J. Mathar, Aug 13 2018
a(n) = A000081(n) - (Sum_{1<=i<=j, i+j=n} A000081(i)*A000081(j)) + (1-(-1)^(n-1)) * binomial(A000081(n/2)+1,2) / 2 [Li, equation 4.2]. - Walt Rorie-Baety, Jul 05 2021

A304867 Number of non-isomorphic hypertrees of weight n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 5, 6, 13, 20, 41, 70, 144, 266, 545, 1072, 2210, 4491, 9388, 19529, 41286, 87361, 186657, 399927, 862584, 1866461, 4058367, 8852686, 19384258, 42570435, 93783472, 207157172, 458805044, 1018564642, 2266475432, 5053991582, 11292781891, 25280844844

Offset: 0

Gus Wiseman, May 20 2018

nonn

A hypertree E is a connected antichain of finite sets satisfying Sum_{e in E} (|e| - 1) = |U(E)| - 1. The weight of a hypertree is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices (see A035053).
From Kevin Ryde, Feb 25 2020: (Start)
a(n), except at n=1, is the number of free trees of n edges (so n+1 vertices) where any two leaves are an even distance apart.  All trees are bipartite graphs and this condition is equivalent to all leaves being in the same bipartite half.  The diameter of a tree is always between two leaves so these trees have even diameter (A000676).
The correspondence between hypertrees and these free trees is described for instance by Bacher (start of section 1.2).  In such a free tree, call a vertex "even" if it is an even distance from a leaf.  The hypertree vertices are these even vertices.  Each hyperedge is the set of vertices surrounding an odd vertex, so hypertree weight is the total number of edges in the free tree.
(End)

			Non-isomorphic representatives of the a(6) = 5 hypertrees are the following:
  {{1,2,3,4,5,6}}
  {{1,2},{1,3,4,5}}
  {{1,2,3},{1,4,5}}
  {{1,2},{1,3},{1,4}}
  {{1,2},{1,3},{2,4}}
Non-isomorphic representatives of the a(7) = 6 hypertrees are the following:
  {{1,2,3,4,5,6,7}}
  {{1,2},{1,3,4,5,6}}
  {{1,2,3},{1,4,5,6}}
  {{1,2},{1,3},{1,4,5}}
  {{1,2},{1,3},{2,4,5}}
  {{1,3},{2,4},{1,2,5}}
From _Kevin Ryde_, Feb 25 2020: (Start)
a(6) = 5 hypertrees of weight 6 and their corresponding free trees of 6 edges (7 vertices).  Each * is an "odd" vertex (odd distance to a leaf).  Each hyperedge is the set of "even" vertices surrounding an odd.
  {1,2,3,4,5,6}       3   2
                       \ /
                      4-*-1      (star 7)
                       / \
                      5   6
  .
  {1,2},{1,3,4,5}               /-3
                      2--*--1--*--4
                                \-5
  .
  {1,2,3},{1,4,5}     2-\       /-4
                         *--1--*
                      3-/       \-5
  .
  {1,2},{1,3},{1,4}    /-*--2
                      1--*--3
                       \-*--4
  .
  {1,2},{2,4},{1,3}   3--*--1--*--2--*--4   (path 7)
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..500
Roland Bacher, On the enumeration of labelled hypertrees and of labelled bipartite trees, arXiv:1102.2708 [math.CO], 2011.

Cf. A007716, A030019, A035053, A048143, A054921, A134955, A134957, A144959, A304911, A304912, A318601.

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b];
EulerT[v_List] := With[{q = etr[v[[#]]&]}, q /@ Range[Length[v]]];
ser[v_] := Sum[v[[i]] x^(i-1), {i, 1, Length[v]}] + O[x]^Length[v];
c[n_] := Module[{v = {1}}, For[i = 1, i <= Ceiling[n/2], i++, v = Join[{1}, EulerT[Join[{0}, EulerT[v]]]]]; v];
seq[n_] := Module[{u = c[n]}, x*ser[EulerT[u]]*(1 - x*ser[u]) + (1 - x)* ser[u] + x + O[x]^n // CoefficientList[#, x]&];
seq[40] (* Jean-François Alcover, Feb 08 2020, after Andrew Howroyd *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
c(n)={my(v=[1]); for(i=1, ceil(n/2), v=concat([1], EulerT(concat([0], EulerT(v))))); v}
seq(n)={my(u=c(n)); Vec(x*Ser(EulerT(u))*(1-x*Ser(u)) + (1 - x)*Ser(u) + x + O(x*x^n))} \\ Andrew Howroyd, Aug 29 2018

a(n) = Sum_{k=1..floor(n/2)} A318601(n+1-k, k). - Andrew Howroyd, Aug 29 2018

Terms a(10) and beyond from Andrew Howroyd, Aug 29 2018

A000677 Number of bicentered trees with n nodes.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 4, 11, 20, 51, 108, 267, 619, 1541, 3762, 9497, 23907, 61216, 157211, 407919, 1063398, 2792026, 7365532, 19535887, 52037837, 139213244, 373820978, 1007420841, 2723783122, 7387129661, 20091790330, 54793762295, 149808274055, 410553630946

Offset: 0

N. J. A. Sloane

See A000676 for more information.

On the bottom of first page 266 of article Cayley (1881) is a table of A000676 and A000677 for n = 1..13. - Michael Somos, Aug 20 2018

			G.f. = x^2 + x^4 + x^5 + 3*x^6 + 4*x^7 + 11*x^8 + 20*x^9 + 51*x^10 + ... - _Michael Somos_, Aug 20 2018

N. L. Biggs et al., Graph Theory 1736-1936, Oxford, 1976, p. 49.
A. Cayley, On the analytical forms called trees, with application to the theory of chemical combinations, Reports British Assoc. Advance. Sci. 45 (1875), 257-305 = Math. Papers, Vol. 9, 427-460 (see p. 438).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Geoffrey Critzer, Table of n, a(n) for n = 0..200 (replacing the first version from N. J. A. Sloane)
Jean-François Alcover, Mathematica program
A. Cayley, On the analytical forms called trees, Amer. J. Math., 4 (1881), 266-268.
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.
J. Riordan, The enumeration of trees by height and diameter, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)
N. J. A. Sloane, Maple program
Eric Weisstein's World of Mathematics, Bicentered Tree.
Index entries for sequences related to trees

Cf. A000055, A000676.

Maple
```
# See link for Maple program.
```
Mathematica
```
(* See link. *)
```

a(n) = A000055(n) - A000676(n).

A027416 Number of unlabeled (and unrooted) trees on n nodes having a centroid.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

nonn

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.

F. Harary, Graph Theory, Addison-Wesley, Reading, MA, 1994; pp. 35, 36.

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.
A. Meir and J. W. Moon, On the branch-sizes of rooted unlabeled trees, in "Graph Theory and Its Applications", Annals New York Acad. Sci., Vol. 576, 1989, pp. 399-407.
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.]
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.)
Index entries for sequences related to rooted trees

Cf. A102911 (trees with a bicentroid), A027415 (trees with a primary branch), A000676 (trees with a center), A000677 (trees with a bicenter), A000055 (trees), A000081 (rooted trees).

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

a(n) = A000055(n) - A102911(n/2) if n is even, else a(n) = A000055(n).
a(n) = A000081(n) - A027415(n). - Emeric Deutsch, Nov 21 2004
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

More terms from Emeric Deutsch, Nov 21 2004

Entry revised (with new definition) by N. J. A. Sloane, Feb 26 2007

cp's OEIS Frontend

A100584 a(n) = A000676(n) - A000677(n).

Views

Author

Keywords

Links

Crossrefs

Formula

A283823 Erroneous version of A000676.

Views

Author

Keywords

Comments

Links

A000055 Number of trees with n unlabeled nodes.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

Python

SageMath

Formula

A304867 Number of non-isomorphic hypertrees of weight n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A000677 Number of bicentered trees with n nodes.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A027416 Number of unlabeled (and unrooted) trees on n nodes having a centroid.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Formula

Extensions

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula