A034799 -id:A034799 - 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

A000243 Number of trees with n nodes, 2 of which are labeled.

Original entry on oeis.org

1, 3, 9, 26, 75, 214, 612, 1747, 4995, 14294, 40967, 117560, 337830, 972027, 2800210, 8075889, 23315775, 67380458, 194901273, 564239262, 1634763697, 4739866803, 13752309730, 39926751310, 115988095896, 337138003197

Offset: 2

N. J. A. Sloane

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).

T. D. Noe, Table of n, a(n) for n=2..200
R. J. Mathar, Topologically distinct sets of non-intersecting circles in the plane, arXiv:1603.00077 [math.CO] (2016), Table 5.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Column k=2 of A034799.

b:= proc(n) option remember; if n<=1 then n else add(k*b(k)* s(n-1, k), k=1..n-1)/(n-1) fi end: s:= proc(n,k) option remember; add(b(n+1-j*k), j=1..iquo(n,k)) end: B:= proc(n) option remember; add(b(k)*x^k, k=1..n) end: a:= n-> coeff(series(B(n-1)^2/(1-B(n-1)), x=0, n+1), x,n): seq(a(n), n=2..27); # Alois P. Heinz, Aug 21 2008

b[n_] := b[n] = If[ n <= 1 , n, Sum[k*b[k]*s[n - 1, k], {k, 1, n - 1}]/(n - 1) ]; s[n_, k_] := s[n, k] = Sum[ b[n + 1 - j*k], {j, 1, Quotient[n, k]}]; B[n_] := B[n] = Sum[ b[k]*x^k, {k, 1, n}]; a[n_] := Coefficient[ Series[ B[n - 1]^2/(1 - B[n - 1]), {x, 0, n + 1}], x, n]; Table[ a[n], {n, 2, 27}] (* Jean-François Alcover, Jan 25 2012, translated from Maple *)

a(n) = A000107(n) - A000081(n). - Christian G. Bower, Nov 15 1999
G.f.: A(x) = B(x)^2/(1-B(x)), where B(x) is g.f. for rooted trees with n nodes, cf. A000081. - Vladeta Jovovic, Oct 19 2001
a(n) = A000106(n) + A304068(n). - Brendan McKay, May 05 2018

More terms and new description from Christian G. Bower, Nov 15 1999

A008295 Triangle read by rows: T(n,k) is the number of partially labeled rooted trees with n vertices, k of which are labeled, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 5, 9, 9, 4, 13, 34, 64, 64, 9, 35, 119, 326, 625, 625, 20, 95, 401, 1433, 4016, 7776, 7776, 48, 262, 1316, 5799, 21256, 60387, 117649, 117649, 115, 727, 4247, 22224, 100407, 373895, 1071904, 2097152, 2097152

Offset: 0

N. J. A. Sloane

T(n, k) where n counts the vertices and 0 <= k <= n counts the labels. - Sean A. Irvine, Mar 22 2018

			Triangle begins with T(0,0):
n\k 0  1   2    3    4    5    6
0   1
1   1  1
2   1  2   2
3   2  5   9    9
4   4 13  34   64   64
5   9 35 119  326  625  625
6  20 95 401 1433 4016 7776 7776

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 134.

Sean A. Irvine, Table of n, a(n) for n = 0..527
J. Riordan, The numbers of labeled colored and chromatic trees, Acta Mathematica 97 (1957) 211.
Index entries for sequences related to rooted trees

Main diagonal is A000169.
Columns k=0..6 are A000081, A000107, A000524, A000444, A000525, A064781, A064782.
Cf. A034799.

m = 9; r[_] = 0;
Do[r[x_] = x Exp[Sum[r[x^k]/k, {k, 1, j}]] + O[x]^j // Normal, {j, 1, m}];
r[x_, y_] = -ProductLog[(-E^(-r[x])) r[x] - (r[x] y)/E^r[x]];
(CoefficientList[#, y] Range[0, Exponent[#, y]]!)& /@ CoefficientList[r[x, y] + O[x]^m, x] /. {} -> {1} // Flatten // Quiet (* Jean-François Alcover, Oct 23 2019 *)

E.g.f.: r(x,y) = T(n,k) * y^k * x^n / k! satisfies r(x,y) * exp(r(x)) = (1+y) * r(x) * exp(r(x,y)) where r(x) is the o.g.f. for A000081. - Sean A. Irvine, Mar 22 2018

More terms from Sean A. Irvine, Mar 22 2018

Name edited by Andrew Howroyd, Mar 23 2023

A000269 Number of trees with n nodes, 3 of which are labeled.

Original entry on oeis.org

3, 16, 67, 251, 888, 3023, 10038, 32722, 105228, 334836, 1056611, 3311784, 10322791, 32026810, 98974177, 304835956, 936147219, 2867586542, 8764280567, 26733395986, 81399821915, 247459136331, 751211286356, 2277496842016

Offset: 3

N. J. A. Sloane

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).

T. D. Noe, Table of n, a(n) for n=3..200
Index entries for sequences related to trees

Column k=3 of A034799.

b[n_] := b[n] = If[n <= 1, n, Sum[k*b[k]*s[n-1, k], {k, 1, n-1}]/(n-1)]; s[n_, k_] := s[n, k] = Sum[b[n+1 - j*k], {j, 1, Quotient[n, k]}]; B[n_] := B[n] = Sum[ b[k]*x^k, {k, 1, n}]; a[n_] := SeriesCoefficient[ B[n-1]^3 * (2*B[n-1]-3) / (B[n-1]-1)^3, {x, 0, n}]; Table[a[n], {n, 3, 30}] (* Jean-François Alcover, Jan 27 2015 *)

G.f.: A(x) = B(x)^3*(3-2*B(x))/(1-B(x))^3, where B(x) is g.f. for rooted trees with n nodes, cf. A000081. - Vladeta Jovovic, Oct 19 2001

a(n) = A000524(n) - 2*A000243(n).

More terms, new description and formula from Christian G. Bower, Nov 15 1999

A000485 Number of partially labeled trees with n nodes (4 of which are labeled).

Original entry on oeis.org

16, 125, 680, 3135, 13155, 51873, 195821, 715614, 2550577, 8911942, 30640888, 103951415, 348724844, 1158722880, 3818514232, 12493703403, 40620949971, 131336770375, 422536529249, 1353341880777, 4317248276746, 13722302173753

Offset: 4

N. J. A. Sloane

nonn

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).

Vincenzo Librandi, Table of n, a(n) for n = 4..1000
Index entries for sequences related to trees

Column k=4 of A034799.

b:= proc(n) option remember; if n<=1 then n else add(k*b(k)* s(n-1, k), k=1..n-1)/(n-1) fi end: s:= proc(n,k) option remember; add(b(n+1-j*k), j=1..iquo(n,k)) end: B:= proc(n) option remember; add(b(k)*x^k, k=1..n) end: a:= n-> coeff(series(B(n-3)^4*(16-19*B(n-3)+6*B(n-3)^2)/(1-B(n-3))^5, x=0, n+1), x,n): seq(a(n), n=4..25); # Alois P. Heinz, Aug 21 2008

b[n_] := b[n] = If[n <= 1, n, Sum[k*b[k]*s[n-1, k], {k, 1, n-1}]/(n-1)]; s[n_, k_] := s[n, k] = Sum[b[n+1-j*k], {j, 1, Quotient[n, k]}]; B[n_] := B[n] = Sum[b[k]*x^k, {k, 1, n}]; a[n_] := SeriesCoefficient[B[n-3]^4*(16-19*B[n-3] + 6*B[n-3]^2)/(1-B[n-3])^5, {x, 0, n}]; Table[a[n], {n, 4, 25}] (* Jean-François Alcover, Mar 20 2014, after Alois P. Heinz *)

G.f.: A(x) = B(x)^4*(16-19*B(x)+6*B(x)^2)/(1-B(x))^5, where B(x) is g.f. for rooted trees with n nodes, cf. A000081.

More terms from Vladeta Jovovic, Oct 19 2001

A000526 Number of partially labeled trees with n nodes (5 of which are labeled).

Original entry on oeis.org

125, 1296, 8716, 47787, 232154, 1040014, 4395772, 17781210, 69498964, 264248924, 982218072, 3582421612, 12857819052, 45515994861, 159205157535, 551049504784, 1889714853263, 6427147635062, 21698583468717

Offset: 5

N. J. A. Sloane

nonn

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).

Vincenzo Librandi, Table of n, a(n) for n = 5..200
Index entries for sequences related to trees

Column k=5 of A034799.

b:= proc(n) option remember; if n<=1 then n else add(k*b(k)* s(n-1, k), k=1..n-1)/(n-1) fi end: s:= proc(n,k) option remember; add(b(n+1-j*k), j=1..iquo(n,k)) end: B:= proc(n) option remember; add(b(k)*x^k, k=1..n) end: a:= n-> coeff(series(B(n-4)^5* (125-204*B(n-4) +118*B(n-4)^2 -24*B(n-4)^3)/ (1-B(n-4))^7, x=0, n+1),x,n): seq(a(n), n=5..23); # Alois P. Heinz, Aug 21 2008

b[n_] := b[n] = If[n <= 1, n, Sum[k*b[k]*s[n-1, k], {k, 1, n-1}]/(n-1)]; s[n_, k_] := s[n, k] = Sum[b[n + 1 - j*k], {j, 1, Quotient[n, k]}]; B[n_] := B[n] = Sum[b[k]*x^k, {k, 1, n}]; a[n_] := SeriesCoefficient[B[n-4]^5*(125 - 204*B[n-4] + 118*B[n-4]^2 - 24*B[n-4]^3)/(1 - B[n-4])^7, {x, 0, n}]; Table[a[n], {n, 5, 23}] (* Jean-François Alcover, Mar 20 2014, after Alois P. Heinz *)

G.f.: A(x) = B(x)^5*(125-204*B(x)+118*B(x)^2-24*B(x)^3)/(1-B(x))^7, where B(x) is g.f. for rooted trees with n nodes, cf. A000081.

More terms from Vladeta Jovovic, Oct 19 2001

A064785 Number of partially labeled trees with n nodes (6 of which are labeled).

Original entry on oeis.org

1296, 16807, 134960, 858578, 4741835, 23786827, 111254536, 493289047, 2096891419, 8614217489, 34402073301, 134162057607, 512703873915, 1925300176534, 7120276125066, 25981116938906, 93678940211218

Offset: 6

Vladeta Jovovic, Oct 19 2001

nonn

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 138.

Column k=6 of A034799.

Cf. A042977.

b:= proc(n) option remember; if n<=1 then n else add(k*b(k)* s(n-1, k), k=1..n-1)/(n-1) fi end: s:= proc(n,k) option remember; add(b(n+1-j*k), j=1..iquo(n,k)) end: B:= proc(n) option remember; add(b(k)*x^k, k=1..n) end: a:= n-> coeff(series(B(n-2)^6* (1296-2633* B(n-2)+ 2128*B(n-2)^2 -806*B(n-2)^3 +120*B(n-2)^4)/ (1-B(n-2))^9, x=0, n+1),x,n): seq(a(n), n=6..22); # Alois P. Heinz, Aug 22 2008

jmax = 23; B[_] = 0;
Do[B[x_] = x*Exp[Sum[B[x^k]/k, {k, 1, j}]]+O[x]^j // Normal, {j, 1, jmax}];
A[x_] = B[x]^6*(1296 - 2633*B[x] + 2128*B[x]^2 - 806*B[x]^3 + 120*B[x]^4)/ (1 - B[x])^9;
CoefficientList[A[x] + O[x]^jmax, x] // Drop[#, 6]& (* Jean-François Alcover, Apr 25 2022 *)

G.f.: A(x) = B(x)^6*(1296-2633*B(x)+2128*B(x)^2-806*B(x)^3+120*B(x)^4)/(1-B(x))^9, where B(x) is g.f. for rooted trees with n nodes, cf. A000081.

A034800 Triangle of number of node labeled trees by number of nodes and number of labels.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 4, 9, 16, 3, 9, 26, 67, 125, 6, 20, 75, 251, 680, 1296, 11, 48, 214, 888, 3135, 8716, 16807, 23, 115, 612, 3023, 13155, 47787, 134960, 262144, 47, 286, 1747, 10038, 51873, 232154, 858578, 2450309, 4782969, 106, 719, 4995, 32722, 195821

Offset: 1

N. J. A. Sloane

Essentially the same as A034799, but omitting the last value from each row. - Sean A. Irvine, Sep 04 2020

			Triangle begins:
  1;
  1, 1;
  1, 2,  3;
  2, 4,  9, 16;
  3, 9, 26, 67, 125;
  ...

J. Riordan, An Introduction to Combinatorial Analysis, p. 138.

Index entries for sequences related to trees

Cf. A034799.

Reference gives generating function.

Offset 1 and more terms from Sean A. Irvine, Sep 04 2020

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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A000243 Number of trees with n nodes, 2 of which are labeled.

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A008295 Triangle read by rows: T(n,k) is the number of partially labeled rooted trees with n vertices, k of which are labeled, 0 <= k <= n.

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A000269 Number of trees with n nodes, 3 of which are labeled.

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A000485 Number of partially labeled trees with n nodes (4 of which are labeled).

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