A002988 -id:A002988 - cp's OEIS frontend

A000081 Number of unlabeled rooted trees with n nodes (or connected functions with a fixed point).

0, 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4766, 12486, 32973, 87811, 235381, 634847, 1721159, 4688676, 12826228, 35221832, 97055181, 268282855, 743724984, 2067174645, 5759636510, 16083734329, 45007066269, 126186554308, 354426847597, 997171512998

Offset: 0

N. J. A. Sloane

Also, number of ways of arranging n-1 nonoverlapping circles: e.g., there are 4 ways to arrange 3 circles, as represented by ((O)), (OO), (O)O, OOO, also see example. (Of course the rules here are different from the usual counting parentheses problems - compare A000108, A001190, A001699.) See Sloane's link for a proof and Vogeler's link for illustration of a(7) as arrangement of 6 circles.
Take a string of n x's and insert n-1 ^'s and n-1 pairs of parentheses in all possible legal ways (cf. A003018). Sequence gives number of distinct functions. The single node tree is "x". Making a node f2 a child of f1 represents f1^f2. Since (f1^f2)^f3 is just f1^(f2*f3) we can think of it as f1 raised to both f2 and f3, that is, f1 with f2 and f3 as children. E.g., for n=4 the distinct functions are ((x^x)^x)^x; (x^(x^x))^x; x^((x^x)^x); x^(x^(x^x)). - W. Edwin Clark and Russ Cox, Apr 29 2003; corrected by Keith Briggs, Nov 14 2005
Also, number of connected multigraphs of order n without cycles except for one loop. - Washington Bomfim, Sep 04 2010
Also, number of planted trees with n+1 nodes.
Also called "Polya trees" by Genitrini (2016). - N. J. A. Sloane, Mar 24 2017

			G.f. = x + x^2 + 2*x^3 + 4*x^4 + 9*x^5 + 20*x^6 + 48*x^7 + 115*x^8 + ...
From _Joerg Arndt_, Jun 29 2014: (Start)
The a(6) = 20 trees with 6 nodes have the following level sequences (with level of root = 0) and parenthesis words:
  01:  [ 0 1 2 3 4 5 ]    (((((())))))
  02:  [ 0 1 2 3 4 4 ]    ((((()()))))
  03:  [ 0 1 2 3 4 3 ]    ((((())())))
  04:  [ 0 1 2 3 4 2 ]    ((((()))()))
  05:  [ 0 1 2 3 4 1 ]    ((((())))())
  06:  [ 0 1 2 3 3 3 ]    (((()()())))
  07:  [ 0 1 2 3 3 2 ]    (((()())()))
  08:  [ 0 1 2 3 3 1 ]    (((()()))())
  09:  [ 0 1 2 3 2 3 ]    (((())(())))
  10:  [ 0 1 2 3 2 2 ]    (((())()()))
  11:  [ 0 1 2 3 2 1 ]    (((())())())
  12:  [ 0 1 2 3 1 2 ]    (((()))(()))
  13:  [ 0 1 2 3 1 1 ]    (((()))()())
  14:  [ 0 1 2 2 2 2 ]    ((()()()()))
  15:  [ 0 1 2 2 2 1 ]    ((()()())())
  16:  [ 0 1 2 2 1 2 ]    ((()())(()))
  17:  [ 0 1 2 2 1 1 ]    ((()())()())
  18:  [ 0 1 2 1 2 1 ]    ((())(())())
  19:  [ 0 1 2 1 1 1 ]    ((())()()())
  20:  [ 0 1 1 1 1 1 ]    (()()()()())
(End)

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N. L. Biggs et al., Graph Theory 1736-1936, Oxford, 1976, pp. 42, 49.
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 305, 998.
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. 451).
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.
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Robert G. Wilson v, Table of n, a(n) for n = 0..1000 (first 201 terms from N. J. A. Sloane)
M. J. H. Al-Kaabi, D. Manchon and F. Patras, Monomial bases and pre-Lie structure for free Lie algebras, arXiv:1708.08312 [math.RA], 2017, See p. 5.
Lluís Alemany-Puig and Ramon Ferrer-i-Cancho, Linear-time calculation of the expected sum of edge lengths in random projective linearizations of trees, arXiv:2107.03277 [cs.CL], 2021.
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David Broadhurst, Resurgent Integer Sequences, Rutgers Experimental Math Seminar, Feb 06 2025; Slides.
A. Cayley, On the analytical forms called trees, Amer. J. Math., 4 (1881), 266-268.
Bartomeu Fiol, Jairo Martínez-Montoya and Alan Rios Fukelman, The planar limit of N=2 superconformal field theories, arXiv:2003.02879 [hep-th], 2020.
P. Flajolet, S. Gerhold and B. Salvy, On the non-holonomic character of logarithms, powers and the n-th prime function, arXiv:math/0501379 [math.CO], 2005.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 71.
Loïc Foissy, Algebraic structures on typed decorated rooted trees, arXiv:1811.07572 [math.RA], 2018.
A. Genitrini, Full asymptotic expansion for Polya structures, arXiv:1605.00837 [math.CO], May 03 2016, p. 6.
Ira M. Gessel, Good Will Hunting's Problem: Counting Homeomorphically Irreducible Trees, arXiv:2305.03157 [math.CO], 2023.
Bernhard Gittenberger, Emma Yu Jin and Michael Wallner, On the shape of random Pólya structures, arXiv|1707.02144 [math.CO], 2017-2018; Discrete Math., 341 (2018), 896-911.
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R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis (annotated cached copy)
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Dominique Manchon, On the mathematics of rooted trees, Université Clermont-Auvergne (France, 2019).
Math Overflow, Discussion
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D. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
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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 10 (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.)
Roger Vogeler, Six Circles, 2015 (illustration for a(7) as the number of arrangements of six circles).
Eric Weisstein's World of Mathematics, Rooted Tree
Eric Weisstein's World of Mathematics, Planted Tree
G. Xiao, Contfrac
Index entries for "core" sequences
Index entries for sequences related to rooted trees
Index entries for sequences related to trees
Index entries for sequences related to parenthesizing
Index entries for continued fractions for constants

Cf. A000041 (partitions), A000055 (unrooted trees), A000169, A001858, A005200, A027750, A051491, A051492, A093637, A187770, A199812, A255170, A087803 (partial sums).
Row sums of A144963. - Gary W. Adamson, Sep 27 2008
Cf. A209397 (log(A(x)/x)).
Cf. A000106 (self-convolution), A002861 (rings of these).
Column k=1 of A033185 and A034799; column k=0 of A008295.

import Data.List (genericIndex)
a000081 = genericIndex a000081_list
a000081_list = 0 : 1 : f 1 [1,0] where
   f x ys = y : f (x + 1) (y : ys) where
     y = sum (zipWith (*) (map h [1..x]) ys) `div` x
     h = sum . map (\d -> d * a000081 d) . a027750_row
-- Reinhard Zumkeller, Jun 17 2013

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; A000081 := [0] cat Eltseq(G); // Geoff Bailey (geoff(AT)maths.usyd.edu.au), Nov 30 2009

N := 30: a := [1,1]; for n from 3 to N do x*mul( (1-x^i)^(-a[i]), i=1..n-1); series(%,x,n+1); b := coeff(%,x,n); a := [op(a),b]; od: a; A000081 := proc(n) if n=0 then 1 else a[n]; fi; end; G000081 := series(add(a[i]*x^i,i=1..N),x,N+2); # also used in A000055
spec := [ T, {T=Prod(Z,Set(T))} ]; A000081 := n-> combstruct[count](spec,size=n); [seq(combstruct[count](spec,size=n), n=0..40)];
# a much more efficient method for computing the result with Maple. It uses two procedures:
a := proc(n) local k; a(n) := add(k*a(k)*s(n-1,k), k=1..n-1)/(n-1) end proc:
a(0) := 0: a(1) := 1: s := proc(n,k) local j; s(n,k) := add(a(n+1-j*k), j=1..iquo(n,k)); # Joe Riel (joer(AT)san.rr.com), Jun 23 2008
# even more efficient, uses the Euler transform:
with(numtheory): a:= proc(n) option remember; local d, j; `if`(n<=1, n, (add(add(d*a(d), d=divisors(j)) *a(n-j), j=1..n-1))/ (n-1)) end:
seq(a(n), n=0..50); # Alois P. Heinz, Sep 06 2008

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 ], {i, 1, 30} ] (* Robert A. Russell *)
a[n_] := a[n] = If[n <= 1, n, Sum[Sum[d*a[d], {d, Divisors[j]}]*a[n-j], {j, 1, n-1}]/(n-1)]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)
a[n_] := a[n] = If[n <= 1, n, Sum[a[n - j] DivisorSum[j, # a[#] &], {j, n - 1}]/(n - 1)]; Table[a[n], {n, 0, 30}] (* Jan Mangaldan, May 07 2014, after Alois P. Heinz *)
(* first do *) << NumericalDifferentialEquationAnalysis`; (* then *)
ButcherTreeCount[30] (* v8 onward Robert G. Wilson v, Sep 16 2014 *)
a[n:0|1] := n; a[n_] := a[n] = Sum[m a[m] a[n-k*m], {m, n-1}, {k, (n-1)/m}]/(n-1); Table[a[n], {n, 0, 30}] (* Vladimir Reshetnikov, Nov 06 2015 *)
terms = 31; A[] = 0; Do[A[x] = x*Exp[Sum[A[x^k]/k, {k, 1, j}]] + O[x]^j // Normal, {j, 1, terms}]; CoefficientList[A[x], x] (* Jean-François Alcover, Jan 11 2018 *)

g(m):= block([si,v],s:0,v:divisors(m), for si in v do (s:s+r(m/si)/si),s);
r(n):=if n=1 then 1 else sum(Co(n-1,k)/k!,k,1,n-1);
Co(n,k):=if k=1  then g(n)  else sum(g(i+1)*Co(n-i-1,k-1),i,0,n-k);
makelist(r(n),n,1,12); /*Vladimir Kruchinin, Jun 15 2012 */

{a(n) = local(A = x); if( n<1, 0, for( k=1, n-1, A /= (1 - x^k + x * O(x^n))^polcoeff(A, k)); polcoeff(A, n))}; /* Michael Somos, Dec 16 2002 */

{a(n) = local(A, A1, an, i); if( n<1, 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]) + polcoeff( if( m%2, A *= (A1 - x^i)^-an[i], A), m-1)); an[n])}; /* Michael Somos, Sep 05 2003 */

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] ) );
concat([0], A) \\ Joerg Arndt, Apr 17 2014

from functools import lru_cache
from sympy import divisors
@lru_cache(maxsize=None)
def divisor_tuple(n): # cached unordered tuple of divisors
    return tuple(divisors(n,generator=True))
@lru_cache(maxsize=None)
def A000081(n): return n if n <= 1 else sum(sum(d*A000081(d) for d in divisor_tuple(k))*A000081(n-k) for k in range(1,n))//(n-1) # Chai Wah Wu, Jan 14 2022

@CachedFunction
def a(n):
    if n < 2: return n
    return add(add(d*a(d) for d in divisors(j))*a(n-j) for j in (1..n-1))/(n-1)
[a(n) for n in range(31)] # Peter Luschny, Jul 18 2014 after Alois P. Heinz

[0]+[RootedTrees(n).cardinality() for n in range(1,31)] # Freddy Barrera, Apr 07 2019

G.f. A(x) satisfies A(x) = x*exp(A(x)+A(x^2)/2+A(x^3)/3+A(x^4)/4+...) [Polya]
Also A(x) = Sum_{n>=1} a(n)*x^n = x / Product_{n>=1} (1-x^n)^a(n).
Recurrence: a(n+1) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} d*a(d) ) * a(n-k+1).
Asymptotically c * d^n * n^(-3/2), where c = A187770 = 0.439924... and d = A051491 = 2.955765... [Polya; Knuth, section 7.2.1.6].
Euler transform is sequence itself with offset -1. - Michael Somos, Dec 16 2001
For n > 1, a(n) = A087803(n) - A087803(n-1). - Vladimir Reshetnikov, Nov 06 2015
For n > 1, a(n) = A123467(n-1). - Falk Hüffner, Nov 26 2015

A000055 Number of trees with n unlabeled nodes.

Original entry on oeis.org

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.
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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.
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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

A000014 Number of series-reduced trees with n nodes.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 2, 2, 4, 5, 10, 14, 26, 42, 78, 132, 249, 445, 842, 1561, 2988, 5671, 10981, 21209, 41472, 81181, 160176, 316749, 629933, 1256070, 2515169, 5049816, 10172638, 20543579, 41602425, 84440886, 171794492, 350238175, 715497037, 1464407113

Offset: 0

N. J. A. Sloane

Other terms for "series-reduced tree": (i) homeomorphically irreducible tree, (ii) homeomorphically reduced tree, (iii) reduced tree, (iv) topological tree.

In a series-reduced tree, vertices cannot have degree 2; they can be leaves or have >= 2 branches.

			G.f. = x + x^2 + x^4 + x^5 + 2*x^6 + 2*x^7 + 4*x^8 + 5*x^9 + 10*x^10 + ...
The star graph with n nodes (except for n=3) is a series-reduced tree. For n=6 the other series-reduced tree is shaped like the letter H. - _Michael Somos_, Dec 19 2014

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 284.
D. G. Cantor, personal communication.
F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 232.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 62, Fig. 3.3.3.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 526.
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).

Matthew Parker, Table of n, a(n) for n = 0..1000 (first 501 terms from Christian G. Bower)
David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], 30 June 2014.
Ira M. Gessel, Good Will Hunting's Problem: Counting Homeomorphically Irreducible Trees, arXiv:2305.03157 [math.CO], 2023.
James Grime and Brady Haran, The problem in Good Will Hunting, 2013 (Numberphile video).
Frank Harary and Geert Prins, The number of homeomorphically irreducible trees and other species, Acta Math., 101 (1959), 141-162.
F. Harary, R. W. Robinson and A. J. Schwenk, Twenty-step algorithm for determining the asymptotic number of trees of various species, J. Austral. Math. Soc., Series A, 20 (1975), 483-503.
F. Harary, R. W. Robinson and A. J. Schwenk, Corrigenda: Twenty-step algorithm for determining the asymptotic number of trees of various species, J. Austral. Math. Soc., Series A 41 (1986), p. 325.
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)
B. D. McKay, Lists of Trees sorted by diameter and Homeomorphically irreducible trees, with <= 22 nodes.
B. D. McKay, Lists of Trees sorted by diameter and Homeomorphically irreducible trees, with <= 22 nodes. [Cached copy of top page only, pdf file, no active links, with permission]
Matthew Parker, The first 2000 terms (7-Zip compressed file)
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 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.)
Eric Weisstein's World of Mathematics, Series-Reduced Tree
Index entries for sequences related to trees
Index entries for "core" sequences

Cf. A000055 (trees), A001678 (series-reduced planted trees), A007827 (series-reduced trees by leaves), A271205 (series-reduced trees by leaves and nodes).

with(powseries): with(combstruct): n := 30: Order := n+3: sys := {B = Prod(C,Z), S = Set(B,1 <= card), C = Union(Z,S)}:
G001678 := (convert(gfseries(sys,unlabeled,x) [S(x)], polynom)) * x^2: G0temp := G001678 + x^2:
G059123 := G0temp / x + G0temp - (G0temp^2+eval(G0temp,x=x^2))/(2*x):
G000014 := ((x-1)/x) * G059123 + ((1+x)/x^2) * G0temp - (1/x^2) * G0temp^2:
A000014 := 0,seq(coeff(G000014,x^i),i=1..n); # Ulrich Schimke (ulrschimke(AT)aol.com)

a[n_] := If[n<1, 0, A = x/(1-x^2) + x*O[x]^n; For[k=3, k <= n-1, k++, A = A/(1 - x^k + x*O[x]^n)^SeriesCoefficient[A, k]]; s = ((Normal[A] /. x -> x^2) + O[x]^(2n))*(1-x) + A*(2-A)*(1+x); SeriesCoefficient[s, n]/2]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 02 2016, adapted from PARI *)

{a(n) = my(A); if( n<1, 0, A = x / (1 - x^2) + x * O(x^n); for(k=3, n-1, A /= (1 - x^k + x * O(x^n))^polcoeff(A, k)); polcoeff( (subst(A, x, x^2) * (1 - x) + A * (2 - A) * (1 + x)) / 2, n))}; /* Michael Somos, Dec 19 2014 */

G.f.: A(x) = ((x-1)/x)*f(x) + ((1+x)/x^2)*g(x) - (1/x^2)*g(x)^2 where f(x) is g.f. for A059123 and g(x) is g.f. for A001678. [Harary and E. M. Palmer, p. 62, Eq. (3.3.10) with extra -(1/x^2)*Hbar(x)^2 term which should be there according to eq.(3.3.14), p. 63, with eq.(3.3.9)]. [corrected by Wolfdieter Lang, Jan 09 2001]

a(n) ~ c * d^n / n^(5/2), where d = A246403 = 2.189461985660850..., c = 0.684447272004914061023163279794145361469033868145768075109924585532604582794... - Vaclav Kotesovec, Aug 25 2014

A002955 Number of (unordered, unlabeled) rooted trimmed trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 17, 36, 79, 175, 395, 899, 2074, 4818, 11291, 26626, 63184, 150691, 361141, 869057, 2099386, 5088769, 12373721, 30173307, 73771453, 180800699, 444101658, 1093104961, 2695730992, 6659914175, 16481146479, 40849449618

Offset: 1

N. J. A. Sloane

A rooted trimmed tree is a tree without limbs of length >= 2. Limbs are the paths from the leafs (towards the root) to the nearest branching point (with the root considered to be a branching point). [clarified by Joerg Arndt, Mar 03 2015]
A rooted tree with a forbidden limb of length k is a rooted tree where the path from any leaf inward hits a branching node or the root within k steps.
Also counts the unordered rooted trees without "x x" in the level sequence for the pre-order walk. The bijection transforms the two outmost nodes in all limbs of lengths >= 2 into V-shaped subtrees. - Joerg Arndt, Mar 03 2015

K. L. McAvaney, personal communication.
A. J. Schwenk, Almost all trees are cospectral, pp. 275-307 of F. Harary, editor, New Directions in the Theory of Graphs. Academic Press, NY, 1973.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (terms n = 1..300 from Vincenzo Librandi)
F. Goebel and R. P. Nederpelt, The number of numerical outcomes of iterated powers, Amer. Math. Monthly, 80 (1971), 1097-1103.
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis (annotated cached copy)
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A002988-A002992, A052318-A052329.

Column k=2 of A255636.

with(numtheory): a:= proc(n) option remember; local d,j,aa; aa:= n-> a(n)-`if`(n=2,1,0); if n<=1 then n else (add(d*aa(d), d=divisors(n-1)) +add(add(d*aa(d), d=divisors(j)) *a(n-j), j=1..n-2))/ (n-1) fi end: seq(a(n), n=1..32); # Alois P. Heinz, Sep 06 2008

a[n_] := a[n] = (Total[ #*b[#]& /@ Divisors[n-1] ] + Sum[ Total[ #*b[#]& /@ Divisors[j] ]*a[n-j], {j, 1, n-2}]) / (n-1); a[1] = 1; b[n_] := a[n]; b[2] = 0; Table[ a[n], {n, 1, 32}](* Jean-François Alcover, Nov 18 2011, after Maple *)

a(n) satisfies a=SHIFT_RIGHT(EULER(a-b)) where b(2)=1, b(k)=0 if k != 2.

a(n) ~ c * d^n / n^(3/2), where d = 2.59952511060090659632378883695107..., c = 0.391083882871301267612387143401... . - Vaclav Kotesovec, Aug 24 2014

More terms, formula and comments from Christian G. Bower, Dec 15 1999

A052318 Number of labeled rooted trimmed trees with n nodes.

Original entry on oeis.org

1, 2, 3, 16, 145, 1536, 19579, 290816, 4942305, 94689280, 2020278931, 47523053568, 1222147737265, 34117226135552, 1027550555918475, 33213871550365696, 1146891651823112641, 42135941698113503232, 1641164216596258397347, 67550839668807638712320

Offset: 1

Christian G. Bower, Dec 15 1999

A rooted trimmed tree is a tree with a forbidden limb of length 2.

A rooted tree with a forbidden limb of length k is a rooted tree where the path from any leaf inward hits a branching node or the root within k steps.

Alois P. Heinz, Table of n, a(n) for n = 1..400
Index entries for sequences related to rooted trees

Cf. A002955, A002988-A002992, A052319-A052329.

A:= proc(n) option remember; if n<=1 then x else convert(series(x* exp(A(n-1)-x^2), x,n), polynom) fi end: a:= n-> coeff(A(n+1), x,n)*n!: seq(a(n), n=1..25); # Alois P. Heinz, Aug 23 2008

a[n_] := Sum[ Boole[ EvenQ[n-m]]*(m^((n+m)/2-2)/((n-m)/2)!)*((-1)^((n-m)/2)/(m-1)!), {m, 1, n}]*n!; Table[a[n], {n, 1, 18}] (* Jean-François Alcover, Sep 10 2012, after Vladimir Kruchinin *)
Rest[CoefficientList[Series[-LambertW[-x/E^(x^2)],{x,0,20}],x]*Range[0,20]!] (* Vaclav Kotesovec, Jan 08 2014 *)

a(n):=sum((if mod(n-m,2)=0 then m^((n+m)/2-2)/((n-m)/2)!*(-1)^((n-m)/2) else 0)/(m-1)!,m,1,n); /* Vladimir Kruchinin, Aug 07 2012 */

E.g.f. satisfies A(x) = x*exp(A(x) - x^2).
E.g.f.: -LambertW(-x/exp(x^2)). - Vaclav Kotesovec, Jan 08 2014
a(n) ~ sqrt(1 + LambertW(-2*exp(-2))) * 2^(n/2) * n^(n-1) / (exp(n) * (-LambertW(-2*exp(-2)))^(n/2)). - Vaclav Kotesovec, Jan 08 2014

A052329 Number of rooted trees with a forbidden limb of length 6.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 47, 113, 281, 706, 1807, 4671, 12224, 32247, 85782, 229683, 618767, 1675618, 4559263, 12457483, 34168574, 94040433, 259637564, 718892281, 1995739380, 5553867981, 15490305017, 43293762352, 121235084565

Offset: 1

Christian G. Bower, Dec 15 1999

nonn

A rooted tree with a forbidden limb of length k is a rooted tree where the path from any leaf inward hits a branching node or the root within k steps.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A002955, A002988-A002992, A052318-A052328.

Column k=6 of A255636.

with(numtheory):
g:= proc(n) g(n):= `if`(n=0, 1, add(add(d*(g(d-1)-
      `if`(d=6, 1, 0)), d=divisors(j))*g(n-j), j=1..n)/n)
    end:
a:= n-> g(n-1):
seq(a(n), n=1..35);  # Alois P. Heinz, Jul 04 2014

g[n_] := g[n] = If[n == 0, 1, Sum[Sum[d*(g[d-1]-If[d == 6, 1, 0]), {d, Divisors[j]} ]*g[n-j], {j, 1, n}]/n]; a[n_] := g[n-1]; Table[a[n], {n, 1, 35}] (* Jean-François Alcover, Feb 24 2015, after Alois P. Heinz *)

a(n) satisfies a=SHIFT_RIGHT(EULER(a-b)) where b(6)=1, b(k)=0 if k != 6.

a(n) ~ c * d^n / n^(3/2), where d = 2.95209316333202396584501452688304..., c = 0.43842619727838455589811980703038... . - Vaclav Kotesovec, Aug 25 2014

A002992 Number of n-node trees with a forbidden limb of length 6.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 6, 10, 22, 45, 102, 226, 531, 1253, 3044, 7456, 18604, 46798, 119133, 305567, 790375, 2057523, 5390759, 14200122, 37598572, 100005401, 267131927, 716318650, 1927758155, 5205240762, 14098580633, 38296720823, 104308468102, 284822276099

Offset: 0

N. J. A. Sloane

nonn

A tree with a forbidden limb of length k is a tree where the path from any leaf inward hits a branching node or another leaf within k steps.

A. J. Schwenk, Almost all trees are cospectral, pp. 275-307 of F. Harary, editor, New Directions in the Theory of Graphs. Academic Press, NY, 1973.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
A. J. Schwenk, Letter to N. J. A. Sloane, Aug 1972
Index entries for sequences related to trees

Cf. A002955, A002988-A002991, A052318-A052329.

with(numtheory):
g:= proc(n) g(n):= `if`(n=0, 1, add(add(d*(g(d-1)-
      `if`(d=6, 1, 0)), d=divisors(j))*g(n-j), j=1..n)/n)
    end:
a:= n-> `if`(n=0, 1, g(n-1)+(`if`(irem(n, 2, 'r')=0,
         g(r-1), 0)-add(g(i-1)*g(n-i-1), i=1..n-1))/2):
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 06 2014

g[n_] := g[n] = If[n == 0, 1, Sum[Sum[d*(g[d-1]-If[d == 6, 1, 0]), {d, Divisors[j] }]*g[n-j], {j, 1, n}]/n]; a[n_] := If[n == 0, 1, g[n-1] + (If[Mod[n, 2 ] == 0, g[Quotient[n, 2]-1], 0] - Sum[g[i-1]*g[n-i-1], {i, 1, n-1}])/2]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 26 2015, after Alois P. Heinz *)

G.f.: 1 + B(x) + (B(x^2) - B(x)^2)/2 where B(x) is g.f. of A052329.

a(n) ~ c * d^n / n^(5/2), where d = 2.95209316333202396584501452688304..., c = 0.52950413787119576841378912289... . - Vaclav Kotesovec, Aug 25 2014

More terms, formula and comments from Christian G. Bower, Dec 15 1999

A000220 Number of asymmetric trees with n nodes (also called identity trees).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 3, 6, 15, 29, 67, 139, 310, 667, 1480, 3244, 7241, 16104, 36192, 81435, 184452, 418870, 955860, 2187664, 5025990, 11580130, 26765230, 62027433, 144133676, 335731381, 783859852, 1834104934, 4300433063, 10102854473, 23778351222

Offset: 1

N. J. A. Sloane

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 330.
S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 301 and 562.
F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 232.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 66, Eq. (3.3.22).
D. E. Knuth, Fundamental Algorithms, 3d Ed. 1997, pp. 386-88 describes methodology for generating similar sequence rapidly.
R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
A. J. Schwenk, personal communication.
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).

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 200 terms from T. D. Noe)
E. Friedman, Illustration of initial terms
F. Harary, R. W. Robinson and A. J. Schwenk, Twenty-step algorithm for determining the asymptotic number of trees of various species, J. Austral. Math. Soc., Series A, 20 (1975), 483-503. Errata: Vol. A 41 (1986), p. 325.
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)
A. J. Schwenk, Letter to N. J. A. Sloane, Aug 1972
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 trees

Cf. A000055, A000081.

Cf. A246169, A004111, A035056.

with(numtheory):
b:= proc(n) option remember; `if`(n<2, n, add(b(n-k)*add(
      b(d)*d*(-1)^(k/d+1), d=divisors(k)), k=1..n-1)/(n-1))
    end:
a:= n-> b(n)-(add(b(j)*b(n-j), j=0..n)+
       `if`(irem(n, 2)=0, b(n/2), 0))/2:
seq(a(n), n=1..50);  # Alois P. Heinz, Feb 24 2015

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 *)

G.f.: A(x)-A^2(x)/2-A(x^2)/2, where A(x) is g.f. for A004111.

a(n) ~ c * d^n / n^(5/2), where d = A246169 = 2.51754035263200389079535..., c = 0.29938828746578432274375484519722721162... . - Vaclav Kotesovec, Aug 25 2014

cp's OEIS Frontend

A000081 Number of unlabeled rooted trees with n nodes (or connected functions with a fixed point).

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A000055 Number of trees with n unlabeled nodes.

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A000014 Number of series-reduced trees with n nodes.

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A002955 Number of (unordered, unlabeled) rooted trimmed trees with n nodes.

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A052318 Number of labeled rooted trimmed trees with n nodes.

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