A331578 -id:A331578 - cp's OEIS frontend

A001678 Number of series-reduced planted trees with n nodes.

0, 0, 1, 0, 1, 1, 2, 3, 6, 10, 19, 35, 67, 127, 248, 482, 952, 1885, 3765, 7546, 15221, 30802, 62620, 127702, 261335, 536278, 1103600, 2276499, 4706985, 9752585, 20247033, 42110393, 87733197, 183074638, 382599946, 800701320, 1677922740, 3520581954

Offset: 0

N. J. A. Sloane

The initial term is 0 by convention, though a good case can be made that it should be 1 instead.
Series-reduced trees contain no node with valency 2; see A000014 for the unrooted series-reduced trees. - Joerg Arndt, Mar 03 2015
For n>=2, a(n+1) is the number of unordered rooted trees (see A000081) with n nodes where nodes cannot have out-degree 1, see example. Imposing the condition only at non-root nodes gives A198518. - Joerg Arndt, Jun 28 2014
For n>=3, a(n+1) is the number of unordered rooted trees with n nodes where all limbs are 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). - Joerg Arndt, Mar 03 2015
A rooted tree is lone-child-avoiding if no vertex has exactly one child, and topologically series-reduced if no vertex has degree 2. This sequence counts unlabeled lone-child-avoiding rooted trees with n - 1 vertices. Topologically series-reduced rooted trees are counted by A001679, which is essentially the same as A059123. - Gus Wiseman, Jan 20 2020

			--------------- Examples (i=internal,e=external): ---------------------------
|.n=2.|..n=4..|..n=5..|...n=6.............|....n=7..........................|
|.....|.......|.......|.............e...e.|................e.e.e......e...e.|
|.....|.e...e.|.e.e.e.|.e.e.e.e...e...i...|.e.e.e.e.e...e....i....e.e...i...|
|..e..|...i...|...i...|....i........i.....|.....i..........i..........i.....|
|..e..|...e...|...e...|....e........e.....|.....e..........e..........e.....|
-----------------------------------------------------------------------------
G.f. = x^2 + x^4 + x^5 + 2*x^6 + 3*x^7 + 6*x^8 + 10*x^9 + 19*x^10 + ...
From _Joerg Arndt_, Jun 28 2014: (Start)
The a(8) = 6 rooted trees with 7 nodes as described in the comment are:
:           level sequence       out-degrees (dots for zeros)
:     1:  [ 0 1 2 3 3 2 1 ]    [ 2 2 2 . . . . ]
:  O--o--o--o
:        .--o
:     .--o
:  .--o
:
:     2:  [ 0 1 2 2 2 2 1 ]    [ 2 4 . . . . . ]
:  O--o--o
:     .--o
:     .--o
:     .--o
:  .--o
:
:     3:  [ 0 1 2 2 2 1 1 ]    [ 3 3 . . . . . ]
:  O--o--o
:     .--o
:     .--o
:  .--o
:  .--o
:
:     4:  [ 0 1 2 2 1 2 2 ]    [ 2 2 . . 2 . . ]
:  O--o--o
:     .--o
:  .--o--o
:     .--o
:
:     5:  [ 0 1 2 2 1 1 1 ]    [ 4 2 . . . . . ]
:  O--o--o
:     .--o
:  .--o
:  .--o
:  .--o
:
:     6:  [ 0 1 1 1 1 1 1 ]    [ 6 . . . . . . ]
:  O--o
:  .--o
:  .--o
:  .--o
:  .--o
:  .--o
:
(End)
From _Gus Wiseman_, Jan 20 2020: (Start)
The a(2) = 1 through a(9) = 10 unlabeled lone-child-avoiding rooted trees with n - 1 nodes (empty n = 3 column shown as dot) are:
  o   .   (oo)  (ooo)  (oooo)   (ooooo)   (oooooo)    (ooooooo)
                       (o(oo))  (o(ooo))  (o(oooo))   (o(ooooo))
                                (oo(oo))  (oo(ooo))   (oo(oooo))
                                          (ooo(oo))   (ooo(ooo))
                                          ((oo)(oo))  (oooo(oo))
                                          (o(o(oo)))  ((oo)(ooo))
                                                      (o(o(ooo)))
                                                      (o(oo)(oo))
                                                      (o(oo(oo)))
                                                      (oo(o(oo)))
(End)

D. G. Cantor, personal communication.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 525.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 62.
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 = 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).
F. Harary and E. M. Palmer, Probability that a point of a tree is fixed, Math. Proc. Camb. Phil. Soc. 85 (1979) 407-415.
F. Harary and G. 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.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 404
Marko Riedel, Generating functions of unordered rooted trees with n nodes where nodes cannot have out-degree 1, classified by the number of leaves, using the Polya Enumeration Theorem and the exponential formula.
Eric Weisstein's World of Mathematics, Series-reduced tree.
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A000014, A007827, A246403, A005202.
Unlabeled rooted trees are counted by A000081.
Topologically series-reduced rooted trees are counted by A001679.
Labeled lone-child-avoiding rooted trees are counted by A060356.
Labeled lone-child-avoiding unrooted trees are counted by A108919.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Singleton-reduced rooted trees are counted by A330951.
Cf. A000669, A004111, A106179, A198518, A254382, A331488, A331578.

with (powseries): with (combstruct): n := 30: sys := {B = Prod(C,Z), S = Set(B,1 <= card), C = Union(Z,S)}: A001678 := 1,0,1,seq(count([S, sys, unlabeled],size=i),i=1..n); # Ulrich Schimke (ulrschimke(AT)aol.com)
# second Maple program:
with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
       d*a(d+1), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n<2, 0,
      `if`(n=2, 1, b(n-2)-a(n-1)))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 02 2014

b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*a[d+1], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]; a[n_] := a[n] = If[n < 2, 0, If[n == 2, 1, b[n-2] - a[n-1]]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Sep 24 2014, after Alois P. Heinz *)
terms = 38; A[] = 0; Do[A[x] = (x^2/(1+x))*Exp[Sum[A[x^k]/(k*x^k), {k, 1, j}]] + O[x]^j // Normal, {j, 1, terms}]; CoefficientList[A[x], x] (* Jean-François Alcover, Jan 12 2018 *)
urt[n_]:=Join@@Table[Union[Sort/@Tuples[urt/@ptn]],{ptn,IntegerPartitions[n-1]}];
Table[If[n<=1,0,Length[Select[urt[n-1],FreeQ[#,{}]&]]],{n,0,10}] (* _Gus Wiseman, Jan 20 2020 *)

(a(n) = if( n<4, n==2, T(n-2, n-3))); /* where */ {T(n, k) = if( n<1 || k<1, (n==0) && (k>=0), sum(j=1, k, sum(i=1, n\j, T(n-i*j, min(n-i*j, j-1)) * binomial( a(j+1) + i-1, i))))}; /* Michael Somos, Jun 04 2002 */

{a(n) = local(A); if( n<3, n==2, A = x / (1 - x^2) + O(x^n); for(k=3, n-2, A /= (1 - x^k + O(x^n))^polcoeff(A, k)); polcoeff(A, n-1))}; /* Michael Somos, Oct 06 2003 */

G.f.: A(x) satisfies A(x) = (x^2/(1+x))*exp( Sum_{k>=1} A(x^k)/(k*x^k) ) [Harary and E. M. Palmer, 1973, p. 62, Eq. (3.3.8)].
G.f.: A(x) = Sum_{n>=2} a(n) * x^n = x^2 / ((1 + x) * Product_{k>0} (1 - x^k)^a(k+1)). - Michael Somos, Oct 06 2003
a(n) ~ c * d^n / n^(3/2), where d = A246403 = 2.189461985660850563... and c = 0.1924225474701550354144525345664845514828912790855223729854471406053655209... - Vaclav Kotesovec, Jun 26 2014
a(n) = Sum_{i=2..n-2} A106179(i, n-1-i) for n >= 3. - Andrew Howroyd, Mar 29 2021

Additional comments from Michael Somos, Jun 05 2002

A001679 Number of series-reduced rooted trees with n nodes.

Original entry on oeis.org

1, 1, 1, 0, 2, 2, 4, 6, 12, 20, 39, 71, 137, 261, 511, 995, 1974, 3915, 7841, 15749, 31835, 64540, 131453, 268498, 550324, 1130899, 2330381, 4813031, 9963288, 20665781, 42947715, 89410092, 186447559, 389397778, 814447067, 1705775653, 3577169927

Offset: 0

N. J. A. Sloane

nonn

Also known as homeomorphically irreducible rooted trees, or rooted trees without nodes of degree 2.

A rooted tree is lone-child-avoiding if no vertex has exactly one child, and topologically series-reduced if no vertex has degree 2. This sequence counts unlabeled topologically series-reduced rooted trees with n vertices. Lone-child-avoiding rooted trees with n - 1 vertices are counted by A001678. - Gus Wiseman, Jan 21 2020

			G.f. = 1 + x + x^2 + 2*x^4 + 2*x^5 + 4*x^6 + 6*x^7 + 12*x^8 + 20*x^9 + ...
From _Gus Wiseman_, Jan 21 2020: (Start)
The a(1) = 1 through a(8) = 12 unlabeled topologically series-reduced rooted trees with n nodes (empty n = 3 column shown as dot) are:
  o  (o)  .  (ooo)   (oooo)   (ooooo)    (oooooo)    (ooooooo)
             ((oo))  ((ooo))  ((oooo))   ((ooooo))   ((oooooo))
                              (oo(oo))   (oo(ooo))   (oo(oooo))
                              ((o(oo)))  (ooo(oo))   (ooo(ooo))
                                         ((o(ooo)))  (oooo(oo))
                                         ((oo(oo)))  ((o(oooo)))
                                                     ((oo(ooo)))
                                                     ((ooo(oo)))
                                                     (o(oo)(oo))
                                                     (oo(o(oo)))
                                                     (((oo)(oo)))
                                                     ((o(o(oo))))
(End)

D. G. Cantor, personal communication.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 62, Eq. (3.3.9).
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).

N. J. A. Sloane, Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
P. J. Cameron, Some treelike objects, Quart. J. Math. Oxford, 38 (1987), 155-183. MR0891613 (89a:05009). See p. 155. - _N. J. A. Sloane_, Apr 18 2014
F. Harary, G. Prins, The number of homeomorphically irreducible trees and other species, Acta Math. 101 (1959) 141-162, W(x,y) equation (9a).
N. J. A. Sloane, Illustration of initial terms
Eric Weisstein's World of Mathematics, Series-Reduced Tree.
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Apart from initial term, same as A059123.
Cf. A000055 (trees by nodes), A000014 (homeomorphically irreducible trees by nodes), A000669 (homeomorphically irreducible planted trees by leaves), A000081 (rooted trees by nodes).
Cf. A246403.
The labeled version is A060313, with unrooted case A005512.
Matula-Goebel numbers of these trees are given by A331489.
Lone-child-avoiding rooted trees are counted by A001678(n + 1).
Cf. A004111, A060356, A198518, A254382, A291636, A330951, A331488, A331578.

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:
G001679 := G0temp / x + G0temp - (G0temp^2+eval(G0temp,x=x^2))/(2*x): A001679 := 0,seq(coeff(G001679,x^i),i=1..n); # Ulrich Schimke (ulrschimke(AT)aol.com)
# adapted for Maple 16 or higher version by Vaclav Kotesovec, Jun 26 2014

terms = 37; (* F = G001678 *) F[] = 0; Do[F[x] = (x^2/(1 + x))*Exp[Sum[ F[x^k]/(k*x^k), {k, 1, j}]] + O[x]^j // Normal, {j, 1, terms + 1}];
G[x_] = 1 + ((1 + x)/x)*F[x] - (F[x]^2 + F[x^2])/(2*x) + O[x]^terms;
CoefficientList[G[x], x] (* Jean-François Alcover, Jan 12 2018 *)
urt[n_]:=Join@@Table[Union[Sort/@Tuples[urt/@ptn]],{ptn,IntegerPartitions[n-1]}];
Table[Length[Select[urt[n],Length[#]!=2&&FreeQ[Z@@#,{}]&]],{n,15}] (* _Gus Wiseman, Jan 21 2020 *)

{a(n) = local(A); if( n<3, n>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( (1 + x)*A - x*(A^2 + subst(A, x, x^2)) / 2, n))};

G.f. = 1 + ((1+x)*f(x) - (f(x)^2+f(x^2))/2)/x where f(x) is g.f. for A001678 (homeomorphically irreducible planted trees by nodes).
a(n) ~ c * d^n / n^(3/2), where d = A246403 = 2.18946198566085056388702757711... and c = 0.4213018528699249210965028... . - Vaclav Kotesovec, Jun 26 2014
For n > 1, this sequence counts lone-child-avoiding rooted trees with n nodes and more than two branches, plus lone-child-avoiding rooted trees with n - 1 nodes. So for n > 1, a(n) = A331488(n) + A001678(n). - Gus Wiseman, Jan 21 2020

Additional comments from Michael Somos, Oct 10 2003

A060313 Number of homeomorphically irreducible rooted trees (also known as series-reduced rooted trees, or rooted trees without nodes of degree 2) on n labeled nodes.

Original entry on oeis.org

1, 2, 0, 16, 25, 576, 2989, 51584, 512649, 8927200, 130956001, 2533847328, 48008533885, 1059817074512, 24196291364925, 609350187214336, 16135860325700881, 459434230368302016, 13788624945433889593, 439102289933675933600, 14705223056221892676741

Offset: 1

Vladeta Jovovic, Mar 27 2001

			From _Gus Wiseman_, Jan 22 2020: (Start)
The a(1) = 1 through a(4) = 16 trees (in the format root[branches], empty column shown as dot) are:
  1  1[2]  .  1[2,3,4]
     2[1]     1[2[3,4]]
              1[3[2,4]]
              1[4[2,3]]
              2[1,3,4]
              2[1[3,4]]
              2[3[1,4]]
              2[4[1,3]]
              3[1,2,4]
              3[1[2,4]]
              3[2[1,4]]
              3[4[1,2]]
              4[1,2,3]
              4[1[2,3]]
              4[2[1,3]]
              4[3[1,2]]
(End)

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.

G. C. Greubel, Table of n, a(n) for n = 1..400
David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
Eric Weisstein's World of Mathematics, Series-reduced tree.
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

The unlabeled unrooted version is A000014.
The unrooted version is A005512.
The unlabeled version is A001679 or A059123.
The lone-child-avoiding version is A060356.
Labeled rooted trees are A000169.
Cf. A000669, A001678, A108919, A254382, A331489, A331578.

[1] cat [n*Factorial(n-2)*(&+[(-1)^k*Binomial(n,k)*(n-k)^(n-k-2)/Factorial(n-k-2): k in [0..n-2]]): n in [2..20]]; // G. C. Greubel, Mar 07 2020

seq( `if`(n=1, 1, n*(n-2)!*add((-1)^k*binomial(n, k)*(n-k)^(n-k-2)/(n-k-2)!, k=0..n-2)), n=1..20); # G. C. Greubel, Mar 07 2020

f[n_] := If[n < 2, 1, n(n - 2)!Sum[(-1)^k*Binomial[n, k](n - k)^(n - 2 - k)/(n - 2 - k)!, {k, 0, n - 2}]]; Table[ f[n], {n, 19}] (* Robert G. Wilson v, Feb 12 2005 *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
lrt[set_]:=If[Length[set]==0,{},Join@@Table[Apply[root,#]&/@Join@@Table[Tuples[lrt/@stn],{stn,sps[DeleteCases[set,root]]}],{root,set}]];
Table[Length[Select[lrt[Range[n]],Length[#]!=2&&FreeQ[Z@@#,Integer[]]&]],{n,6}] (* Gus Wiseman, Jan 22 2020 *)

[1]+[n*factorial(n-2)*sum((-1)^k*binomial(n,k)*(n-k)^(n-k-2)/factorial( n-k-2) for k in (0..n-2)) for n in (2..20)] # G. C. Greubel, Mar 07 2020

a(n) = n*(n-2)!*Sum_{k=0..n-2} (-1)^k*binomial(n, k)*(n-k)^(n-k-2)/(n-k-2)!, n>1.
E.g.f.: x*(exp( - LambertW(-x/(1+x))) - (LambertW(-x/(1+x))/2 )^2).
a(n) ~ n^(n-1) * (1-exp(-1))^(n+1/2). - Vaclav Kotesovec, Oct 05 2013
E.g.f.: -(1+x)*LambertW(-x/(1+x)) - (x/2)*LambertW(-x/(1+x))^2. - G. C. Greubel, Mar 07 2020

A254382 Number of rooted labeled trees on n nodes such that every nonroot node is the child of a branching node or of the root.

Original entry on oeis.org

0, 1, 2, 3, 16, 85, 696, 6349, 72080, 918873, 13484080, 219335281, 3962458248, 78203547877, 1680235050872, 38958029188485, 970681471597216, 25847378934429361, 732794687650764000, 22032916968153975769, 700360446794528578520

Offset: 0

Geoffrey Critzer, Jan 29 2015

nonn

Here, a branching node is a node with at least two children.
In other words, a(n) is the number of labeled rooted trees on n nodes such that the path from every node towards the root reaches a branching node (or the root) in one step.
Also labeled rooted trees that are lone-child-avoiding except possibly for the root. The unlabeled version is A198518. - Gus Wiseman, Jan 22 2020

			a(5) = 85:
...0................0...............0-o...
...|.............../ \............ /|\....
...o..............o   o...........o o o...
../|\............/ \   ...................
.o o o..........o   o   ..................
These trees have 20 + 60 + 5 = 85 labelings.
From _Gus Wiseman_, Jan 22 2020: (Start)
The a(1) = 1 through a(4) = 16 trees (in the format root[branches]) are:
  1  1[2]  1[2,3]  1[2,3,4]
     2[1]  2[1,3]  1[2[3,4]]
           3[1,2]  1[3[2,4]]
                   1[4[2,3]]
                   2[1,3,4]
                   2[1[3,4]]
                   2[3[1,4]]
                   2[4[1,3]]
                   3[1,2,4]
                   3[1[2,4]]
                   3[2[1,4]]
                   3[4[1,2]]
                   4[1,2,3]
                   4[1[2,3]]
                   4[2[1,3]]
                   4[3[1,2]]
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..200
David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Cf. A231797, A052318 (condition is applied only to leaf nodes).
The unlabeled version is A198518
The non-planted case is A060356.
Labeled rooted trees are A000169.
Lone-child-avoiding rooted trees are A001678(n + 1).
Labeled topologically series-reduced rooted trees are A060313.
Labeled lone-child-avoiding unrooted trees are A108919.
Cf. A000014, A000669, A001679, A005512, A291636, A331488, A331578.

nn = 20; b = 1 + Sum[nn = n; n! Coefficient[Series[(Exp[x] - x)^n, {x, 0, nn}], x^n]*x^n/n!, {n,1, nn}]; c = Sum[a[n] x^n/n!, {n, 0, nn}]; sol = SolveAlways[b == Series[1/(1 - (c - x)), {x, 0, nn}], x]; Flatten[Table[a[n], {n, 0, nn}] /. sol]
nn = 30; CoefficientList[Series[1+x-1/Sum[SeriesCoefficient[(E^x-x)^n,{x,0,n}]*x^n,{n,0,nn}],{x,0,nn}],x] * Range[0,nn]! (* Vaclav Kotesovec, Jan 30 2015 *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
lrt[set_]:=If[Length[set]==0,{},Join@@Table[Apply[root,#]&/@Join@@Table[Tuples[lrt/@stn],{stn,sps[DeleteCases[set,root]]}],{root,set}]];
Table[Length[Select[lrt[Range[n]],FreeQ[Z@@#,Integer[]]&]],{n,6}] (* Gus Wiseman, Jan 22 2020 *)

E.g.f.: A(x) satisfies 1/(1 - (A(x) - x)) = B(x) where B(x) is the e.g.f. for A231797.

a(n) ~ (1-exp(-1))^(n-1/2) * n^(n-1). - Vaclav Kotesovec, Jan 30 2015

A331233 Number of unlabeled rooted trees with n vertices and more than two branches of the root.

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 12, 30, 75, 194, 501, 1317, 3485, 9302, 24976, 67500, 183290, 500094, 1369939, 3766831, 10391722, 28756022, 79794407, 221987348, 619019808, 1729924110, 4844242273, 13590663071, 38195831829, 107523305566, 303148601795, 855922155734, 2419923253795

Offset: 1

Gus Wiseman, Jan 21 2020

nonn

			The a(4) = 1 through a(7) = 12 rooted trees:
  (ooo)  (oooo)   (ooooo)    (oooooo)
         (oo(o))  (oo(oo))   (oo(ooo))
                  (ooo(o))   (ooo(oo))
                  (o(o)(o))  (oooo(o))
                  (oo((o)))  (o(o)(oo))
                             (oo((oo)))
                             (oo(o)(o))
                             (oo(o(o)))
                             (ooo((o)))
                             ((o)(o)(o))
                             (o(o)((o)))
                             (oo(((o))))

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 500 terms from Andrew Howroyd)

The Matula-Goebel numbers of these trees are given by A033942.
The series-reduced case is A331488.
The lone-child-avoiding case is (also) A331488.
The labeled version is A331577.
Unlabeled rooted trees are counted by A000081.
Cf. A001678, A001679, A004111, A033185, A060313, A206429, A331490, A331578.

g:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
      `if`(i<1, 0, add(binomial(g(i-1$2, 0)+j-1, j)*
         g(n-i*j, i-1, max(0, t-j)), j=0..n/i)))
    end:
a:= n-> g(n-1$2, 3):
seq(a(n), n=1..40);  # Alois P. Heinz, Jan 22 2020

urt[n_]:=Join@@Table[Union[Sort/@Tuples[urt/@ptn]],{ptn,IntegerPartitions[n-1]}];
Table[Length[Select[urt[n],Length[#]>2&]],{n,10}]
(* Second program: *)
g[n_, i_, t_] := g[n, i, t] = If[n == 0, If[t == 0, 1, 0],
     If[i < 1, 0, Sum[Binomial[g[i - 1, i - 1, 0] + j - 1, j]*
     g[n - i*j, i - 1, Max[0, t - j]], {j, 0, n/i}]]];
a[n_] := g[n-1, n-1, 3];
Array[a, 40] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)

\\ TreeGf gives gf of A000081.
TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
seq(n)={my(g=TreeGf(n)); Vec(g - x*(1 + g + (g^2 + subst(g, x, x^2))/2), -n)} \\ Andrew Howroyd, Jan 22 2020

For n > 1, a(n) = Sum_{k > 2} A033185(n - 1, k).

G.f.: f(x) - x*(1 + f(x) + (f(x)^2 + f(x^2))/2) where f(x) is the g.f. of A000081. - Andrew Howroyd, Jan 22 2020

A331577 Number of labeled rooted trees with n vertices and more than two branches of the root.

Original entry on oeis.org

0, 0, 0, 4, 65, 1026, 17857, 349224, 7657281, 186895270, 5037424601, 148805552556, 4784793219505, 166458635341194, 6231891513395745, 249886992888096976, 10686839817678846209, 485632267141865950926, 23370062118676064101801, 1187393725239246382405140

Offset: 1

Gus Wiseman, Jan 21 2020

nonn

			Non-isomorphic representatives of the a(6) = 1026 trees (in the format root[branches]) are:
  1[2,3,4[5[6]]]
  1[2,3[4],5[6]]
  1[2,3,4[5,6]]
  1[2,3,4,5[6]]
  1[2,3,4,5,6]

Andrew Howroyd, Table of n, a(n) for n = 1..200

The series-reduced version is A331578.
The unlabeled version is A331233.
Labeled rooted trees are counted by A000169.
Cf. A000081, A033185, A060313, A060356, A206429, A331488, A331490.

lrt[set_]:=If[Length[set]==0,{},Join@@Table[Apply[root,#]&/@Join@@Table[Tuples[lrt/@stn],{stn,sps[DeleteCases[set,root]]}],{root,set}]];
Table[Length[Select[lrt[Range[n]],Length[#]>2&]],{n,6}]

seq(n)={my(f=serreverse(x*exp(O(x^n) -x ))); Vec(serlaplace(f - x*(1 + f + f^2/2)), -n)} \\ Andrew Howroyd, Jan 23 2020

For n > 1, a(n) = Sum_{k > 2} A206429(n, k).

E.g.f.: f(x) - x*(1 + f(x) + f(x)^2/2), where f(x) is the e.g.f. of A000169. - Andrew Howroyd, Jan 23 2020

cp's OEIS Frontend

A001678 Number of series-reduced planted trees with n nodes.

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A001679 Number of series-reduced rooted trees with n nodes.

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A060313 Number of homeomorphically irreducible rooted trees (also known as series-reduced rooted trees, or rooted trees without nodes of degree 2) on n labeled nodes.

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A254382 Number of rooted labeled trees on n nodes such that every nonroot node is the child of a branching node or of the root.

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A331233 Number of unlabeled rooted trees with n vertices and more than two branches of the root.

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A331577 Number of labeled rooted trees with n vertices and more than two branches of the root.

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