A319312 -id:A319312 - cp's OEIS frontend

A339645 Triangle read by rows: T(n,k) is the number of inequivalent colorings of lone-child-avoiding rooted trees with n colored leaves using exactly k colors.

1, 1, 1, 2, 3, 2, 5, 17, 12, 5, 12, 73, 95, 44, 12, 33, 369, 721, 512, 168, 33, 90, 1795, 5487, 5480, 2556, 625, 90, 261, 9192, 41945, 58990, 36711, 12306, 2342, 261, 766, 47324, 321951, 625088, 516952, 224241, 57155, 8702, 766, 2312, 249164, 2483192, 6593103, 7141755, 3965673, 1283624, 258887, 32313, 2312

Offset: 1

Andrew Howroyd, Dec 11 2020

Only the leaves are colored. Equivalence is up to permutation of the colors.

Lone-child-avoiding rooted trees are also called planted series-reduced trees in some other sequences.

			Triangle begins:
    1;
    1,     1;
    2,     3,      2;
    5,    17,     12,      5;
   12,    73,     95,     44,     12;
   33,   369,    721,    512,    168,     33;
   90,  1795,   5487,   5480,   2556,    625,    90;
  261,  9192,  41945,  58990,  36711,  12306,  2342,  261;
  766, 47324, 321951, 625088, 516952, 224241, 57155, 8702, 766;
  ...
From _Gus Wiseman_, Jan 02 2021: (Start)
Non-isomorphic representatives of the 39 = 5 + 17 + 12 + 5 trees with four colored leaves:
  (1111)      (1112)      (1123)      (1234)
  (1(111))    (1122)      (1(123))    (1(234))
  (11(11))    (1(112))    (11(23))    (12(34))
  ((11)(11))  (11(12))    (12(13))    ((12)(34))
  (1(1(11)))  (1(122))    (2(113))    (1(2(34)))
              (11(22))    (23(11))
              (12(11))    ((11)(23))
              (12(12))    (1(1(23)))
              (2(111))    ((12)(13))
              ((11)(12))  (1(2(13)))
              (1(1(12)))  (2(1(13)))
              ((11)(22))  (2(3(11)))
              (1(1(22)))
              (1(2(11)))
              ((12)(12))
              (1(2(12)))
              (2(1(11)))
(End)

Andrew Howroyd, PARI Functions for Combinatorial Species, v2, Dec 2020.
Wikipedia, Combinatorial species

The case with only one color is A000669.
Counting by nodes gives A318231.
A labeled version is A319376.
Row sums are A330470.
A000311 counts singleton-reduced phylogenetic trees.
A001678 counts unlabeled lone-child-avoiding rooted trees.
A005121 counts chains of set partitions, with maximal case A002846.
A005804 counts phylogenetic rooted trees with n labels.
A060356 counts labeled lone-child-avoiding rooted trees.
A141268 counts lone-child-avoiding rooted trees with leaves summing to n.
A291636 lists Matula-Goebel numbers of lone-child-avoiding rooted trees.
A316651 counts lone-child-avoiding rooted trees with normal leaves.
A316652 counts lone-child-avoiding rooted trees with strongly normal leaves.
A330465 counts inequivalent leaf-colorings of phylogenetic rooted trees.
Cf. A196545, A213427, A281118, A289501, A292504, A318812, A319312, A330627.

\\ See link above for combinatorial species functions.
cycleIndexSeries(n)={my(v=vector(n)); v[1]=sv(1); for(n=2, #v, v[n] = polcoef( sExp(x*Ser(v[1..n])), n )); x*Ser(v)}
{my(A=InequivalentColoringsTriangle(cycleIndexSeries(10))); for(n=1, #A~, print(A[n,1..n]))}

A300660 Number of unlabeled rooted phylogenetic trees with n (leaf-) nodes such that for each inner node all children are either leaves or roots of distinct subtrees.

Original entry on oeis.org

0, 1, 1, 2, 3, 6, 13, 30, 72, 182, 467, 1222, 3245, 8722, 23663, 64758, 178459, 494922, 1380105, 3867414, 10884821, 30756410, 87215419, 248117618, 707952902, 2025479210, 5809424605, 16700811214, 48113496645, 138884979562, 401645917999, 1163530868090

Offset: 0

Alois P. Heinz, Jun 18 2018

From Gus Wiseman, Jul 31 2018 and Feb 06 2020: (Start)
a(n) is the number of lone-child-avoiding rooted identity trees whose leaves form an integer partition of n. For example, the following are the a(6) = 13 lone-child-avoiding rooted identity trees whose leaves form an integer partition of 6.
  6,
  (15),
  (24),
  (123), (1(23)), (2(13)), (3(12)),
  (1(14)),
  (1(1(13))),
  (12(12)), (1(2(12))), (2(1(12))),
  (1(1(1(12)))).
(End)

			:   a(3) = 2:        :   a(4) = 3:                      :
:      o       o     :        o         o        o      :
:     / \     /|\    :       / \       / \     /( )\    :
:    o   N   N N N   :      o   N     o   N   N N N N   :
:   ( )              :     / \       /|\                :
:   N N              :    o   N     N N N               :
:                    :   ( )                            :
:                    :   N N                            :
From _Gus Wiseman_, Feb 06 2020: (Start)
The a(2) = 1 through a(6) = 13 unlabeled rooted phylogenetic semi-identity trees:
  (oo) (ooo)     (oooo)         (ooooo)             (oooooo)
       ((o)(oo)) ((o)(ooo))     ((o)(oooo))         ((o)(ooooo))
                 ((o)((o)(oo))) ((oo)(ooo))         ((oo)(oooo))
                                ((o)((o)(ooo)))     ((o)(oo)(ooo))
                                ((oo)((o)(oo)))     (((o)(oo))(ooo))
                                ((o)((o)((o)(oo)))) ((o)((o)(oooo)))
                                                    ((o)((oo)(ooo)))
                                                    ((oo)((o)(ooo)))
                                                    ((o)(oo)((o)(oo)))
                                                    ((o)((o)((o)(ooo))))
                                                    ((o)((oo)((o)(oo))))
                                                    ((oo)((o)((o)(oo))))
                                                    ((o)((o)((o)((o)(oo)))))
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..2079
Wikipedia, Phylogenetic 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. A000081, A004111, A141268, A289501, A301467.
Cf. A000669, A001678, A005804, A292504, A300660, A316653, A316654, A316656.
The locally disjoint case is A316694.
Cf. A276625, A306200, A319312, A331679, A331686, A331875.

b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial(a(i), j), j=0..n/i)))
    end:
a:= n-> `if`(n=0, 0, 1+b(n, n-1)):
seq(a(n), n=0..30);

b[0, ] = 1; b[, _?NonPositive] = 0;
b[n_, i_] := b[n, i] = Sum[b[n-i*j, i-1]*Binomial[a[i], j], {j, 0, n/i}];
a[0] = 0; a[n_] := a[n] = 1 + b[n, n-1];
Table[a[n], {n, 0, 31}] (* Jean-François Alcover, May 03 2019, from Maple *)
ursit[n_]:=Prepend[Join@@Table[Select[Union[Sort/@Tuples[ursit/@ptn]],UnsameQ@@#&],{ptn,Select[IntegerPartitions[n],Length[#]>1&]}],n];
Table[Length[ursit[n]],{n,10}] (* Gus Wiseman, Feb 06 2020 *)

a(n) ~ c * d^n / n^(3/2), where d = 3.045141208159736483720243229947630323380565686... and c = 0.2004129296838557718008171812000512670126... - Vaclav Kotesovec, Aug 27 2018

A320160 Number of series-reduced balanced rooted trees whose leaves form an integer partition of n.

Original entry on oeis.org

1, 2, 3, 6, 9, 19, 31, 63, 110, 215, 391, 773, 1451, 2879, 5594, 11173, 22041, 44136, 87631, 175155, 348186, 694013, 1378911, 2743955, 5452833, 10853541, 21610732, 43122952, 86192274, 172753293, 347114772, 699602332, 1414033078, 2866580670, 5826842877, 11874508385

Offset: 1

Gus Wiseman, Oct 06 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches, and balanced if all leaves are the same distance from the root.

Also the number of balanced unlabeled phylogenetic rooted trees with n leaves.

			The a(1) = 1 through a(6) = 19 rooted trees:
  1  2     3      4           5            6
     (11)  (12)   (13)        (14)         (15)
           (111)  (22)        (23)         (24)
                  (112)       (113)        (33)
                  (1111)      (122)        (114)
                  ((11)(11))  (1112)       (123)
                              (11111)      (222)
                              ((11)(12))   (1113)
                              ((11)(111))  (1122)
                                           (11112)
                                           (111111)
                                           ((11)(13))
                                           ((11)(22))
                                           ((12)(12))
                                           ((11)(112))
                                           ((12)(111))
                                           ((11)(1111))
                                           ((111)(111))
                                           ((11)(11)(11))

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A000081, A000311, A000669, A001678, A005804, A048816, A079500, A119262, A120803, A141268, A244925, A319312.

Cf. A316624, A320154, A320155, A320169, A320173, A320179.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
phy2[labs_]:=If[Length[labs]==1,labs,Union@@Table[Sort/@Tuples[phy2/@ptn],{ptn,Select[mps[Sort[labs]],Length[#1]>1&]}]];
Table[Sum[Length[Select[phy2[ptn],SameQ@@Length/@Position[#,_Integer]&]],{ptn,IntegerPartitions[n]}],{n,8}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(u=vector(n, n, 1), v=vector(n)); while(u, v+=u; u=EulerT(u)-u); v} \\ Andrew Howroyd, Oct 25 2018

Terms a(14) and beyond from Andrew Howroyd, Oct 25 2018

A330465 Number of non-isomorphic series-reduced rooted trees whose leaves are multisets with a total of n elements.

Original entry on oeis.org

1, 4, 14, 87, 608, 5573, 57876, 687938, 9058892, 130851823, 2048654450, 34488422057, 620046639452, 11839393796270, 238984150459124, 5079583100918338, 113299159314626360, 2644085918303683758, 64393240540265515110, 1632731130253043991252, 43013015553755764179000

Offset: 1

Gus Wiseman, Dec 21 2019

nonn

Also inequivalent leaf-colorings of phylogenetic rooted trees with n labels. A phylogenetic rooted tree is a series-reduced rooted tree whose leaves are (usually disjoint) sets.

			Non-isomorphic representatives of the a(3) = 14 trees:
  ((1)((1)(1)))  ((1)((1)(2)))  ((1)((2)(3)))  ((2)((1)(1)))
  ((1)(1)(1))    ((1)(1)(2))    ((1)(2)(3))    ((2)(1,1))
  ((1)(1,1))     ((1)(1,2))     ((1)(2,3))
  (1,1,1)        (1,1,2)        (1,2,3)

The version where leaves are atoms is A318231.
The case with sets as leaves is A330624.
The case with disjoint sets as leaves is A141268.
Labeled versions are A330467 (strongly normal) and A330469 (normal).
The singleton-reduced version is A330470.
Cf. A000311, A000669, A004114, A005804, A007716, A281118, A289501, A292504, A316651, A316652, A318812, A319312, A330471, A330474, A330625, A339645.

\\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(v=vector(n), p=sEulerT(x*sv(1) + O(x*x^n))); v[1]=sv(1); for(n=2, #v, v[n] = polcoef( sEulerT(x*Ser(v[1..n])), n ) + polcoef(p,n)); x*Ser(v)}
InequivalentColoringsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 13 2020

Terms a(7) and beyond from Andrew Howroyd, Dec 13 2020

A320154 Number of series-reduced balanced rooted trees whose leaves form a set partition of {1,...,n}.

Original entry on oeis.org

1, 2, 5, 18, 92, 588, 4328, 35920, 338437, 3654751, 45105744, 625582147, 9539374171, 157031052142, 2757275781918, 51293875591794, 1007329489077804, 20840741773898303, 453654220906310222, 10380640686263467204, 249559854371799622350, 6301679967177242849680

Offset: 1

Gus Wiseman, Oct 06 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches, and balanced if all leaves are the same distance from the root.

Also the number of balanced phylogenetic rooted trees on n distinct labels.

			The a(1) = 1 through a(4) = 18 rooted trees:
  (1)  (12)      (123)        (1234)
       ((1)(2))  ((1)(23))    ((1)(234))
                 ((2)(13))    ((12)(34))
                 ((3)(12))    ((13)(24))
                 ((1)(2)(3))  ((14)(23))
                              ((2)(134))
                              ((3)(124))
                              ((4)(123))
                              ((1)(2)(34))
                              ((1)(3)(24))
                              ((1)(4)(23))
                              ((2)(3)(14))
                              ((2)(4)(13))
                              ((3)(4)(12))
                              ((1)(2)(3)(4))
                              (((1)(2))((3)(4)))
                              (((1)(3))((2)(4)))
                              (((1)(4))((2)(3)))

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

Cf. A000081, A000311, A000669, A001678, A005804, A048816, A079500, A119262, A120803, A141268, A244925, A292504, A300660, A319312.

Cf. A320155, A320160, A316624, A320169, A320173, A320176, A320179.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
gug[m_]:=Prepend[Join@@Table[Union[Sort/@Tuples[gug/@mtn]],{mtn,Select[sps[m],Length[#]>1&]}],m];
Table[Length[Select[gug[Range[n]],SameQ@@Length/@Position[#,_Integer]&]],{n,9}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
b(n,k)={my(u=vector(n), v=vector(n)); u[1]=k; u=EulerT(u); while(u, v+=u; u=EulerT(u)-u); v}
seq(n)={my(M=Mat(vectorv(n,k,b(n,k)))); vector(n, k, sum(i=1, k, binomial(k,i)*(-1)^(k-i)*M[i,k]))} \\ Andrew Howroyd, Oct 26 2018

Terms a(9) and beyond from Andrew Howroyd, Oct 26 2018

A331686 Number of lone-child-avoiding locally disjoint rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 2, 4, 8, 17, 41, 103, 280, 793, 2330, 6979, 21291

Offset: 1

Gus Wiseman, Jan 31 2020

A rooted tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other (unequal) child of the same vertex. Lone-child-avoiding means there are no unary branchings. In an identity tree, all branches of any given vertex are distinct.

			The a(1) = 1 through a(5) = 17 trees:
  (1)  (2)   (3)       (4)            (5)
       (11)  (12)      (13)           (14)
             (111)     (22)           (23)
             ((1)(2))  (112)          (113)
                       (1111)         (122)
                       ((1)(3))       (1112)
                       ((2)(11))      (11111)
                       ((1)((1)(2)))  ((1)(4))
                                      ((2)(3))
                                      ((1)(22))
                                      ((3)(11))
                                      ((2)(111))
                                      ((1)((1)(3)))
                                      ((2)((1)(2)))
                                      ((11)((1)(2)))
                                      ((1)((2)(11)))
                                      ((1)((1)((1)(2))))

The non-identity version is A331678.
The case where the leaves are all singletons is A316694.
Identity trees are A004111.
Locally disjoint identity trees are A316471.
Locally disjoint enriched identity p-trees are A331684.
Lone-child-avoiding locally disjoint rooted semi-identity trees are A212804.
Cf. A000669, A001678, A005804, A141268, A300660, A316697, A319312, A331679, A331683, A331783, A331874, A331875.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
mpti[m_]:=Prepend[Join@@Table[Select[Union[Sort/@Tuples[mpti/@p]],UnsameQ@@#&&disjointQ[#]&],{p,Select[mps[m],Length[#]>1&]}],m];
Table[Sum[Length[mpti[m]],{m,Sort/@IntegerPartitions[n]}],{n,8}]

A316624 Number of balanced p-trees with n leaves.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 8, 9, 16, 20, 40, 47, 83, 111, 201, 259, 454, 603, 1049, 1432, 2407, 3390, 6006, 8222, 13904, 20304, 34828, 50291, 85817, 126013, 217653, 317894, 535103, 798184, 1367585, 2008125, 3360067, 5048274, 8499942, 12623978, 21023718, 31552560, 52575257

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

A p-tree of weight n is either a single node (if n = 1) or a finite sequence of p-trees whose weights are weakly decreasing and sum to n.

A tree is balanced if all leaves have the same height.

			The a(1) = 1 through a(7) = 4 balanced p-trees:
  o  (oo)  (ooo)  (oooo)      (ooooo)      (oooooo)        (ooooooo)
                  ((oo)(oo))  ((ooo)(oo))  ((ooo)(ooo))    ((oooo)(ooo))
                                           ((oooo)(oo))    ((ooooo)(oo))
                                           ((oo)(oo)(oo))  ((ooo)(oo)(oo))

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A000311, A000669, A001678, A005804, A048816, A079500, A119262, A120803, A141268, A196545, A289501, A319312.

ptrs[n_]:=If[n==1,{"o"},Join@@Table[Tuples[ptrs/@p],{p,Rest[IntegerPartitions[n]]}]];
Table[Length[ptrs[n]],{n,12}]
Table[Length[Select[ptrs[n],SameQ@@Length/@Position[#,"o"]&]],{n,12}]

seq(n)={my(p=x + O(x*x^n), q=0); while(p, q+=p; p = 1/prod(k=1, n, 1 - polcoef(p,k)*x^k + O(x*x^n)) - 1 - p); Vec(q)} \\ Andrew Howroyd, Oct 26 2018

Terms a(17) and beyond from Andrew Howroyd, Oct 26 2018

A320328 Number of square multiset partitions of integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 6, 11, 20, 36, 65, 117, 214, 382, 679

Offset: 0

Gus Wiseman, Oct 11 2018

A multiset partition is square if its length is equal to its number of distinct atoms.

			The a(1) = 1 through a(6) = 20 square partitions:
  {{1}}  {{2}}    {{3}}      {{4}}        {{5}}          {{6}}
         {{1,1}}  {{1,1,1}}  {{2,2}}      {{1},{4}}      {{3,3}}
                  {{1},{2}}  {{1},{3}}    {{2},{3}}      {{1},{5}}
                             {{1,1,1,1}}  {{1},{1,3}}    {{2,2,2}}
                             {{1},{1,2}}  {{1},{2,2}}    {{2},{4}}
                             {{2},{1,1}}  {{2},{1,2}}    {{1},{1,4}}
                                          {{3},{1,1}}    {{4},{1,1}}
                                          {{1,1,1,1,1}}  {{1},{1,1,3}}
                                          {{1},{1,1,2}}  {{1,1},{1,3}}
                                          {{1,1},{1,2}}  {{1},{1,2,2}}
                                          {{2},{1,1,1}}  {{1,1},{2,2}}
                                                         {{1,2},{1,2}}
                                                         {{1},{2},{3}}
                                                         {{2},{1,1,2}}
                                                         {{3},{1,1,1}}
                                                         {{1,1,1,1,1,1}}
                                                         {{1},{1,1,1,2}}
                                                         {{1,1},{1,1,2}}
                                                         {{1,2},{1,1,1}}
                                                         {{2},{1,1,1,1}}

Cf. A000219, A001970, A047968, A050342, A089259, A141268, A261049, A289501, A305551, A316983, A319066, A319312, A319616, A320330, A320331.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],Length[#]==Length[Union@@#]&]],{n,8}]

A331679 Number of lone-child-avoiding locally disjoint rooted trees whose leaves are positive integers summing to n, with no two distinct leaves directly under the same vertex.

Original entry on oeis.org

1, 2, 3, 8, 16, 48, 116, 341, 928, 2753, 7996, 24254, 73325, 226471, 702122

Offset: 1

Gus Wiseman, Jan 25 2020

A tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other unequal child of the same vertex. It is lone-child-avoiding if there are no unary branchings.

			The a(1) = 1 through a(5) = 16 trees:
  1  2     3        4           5
     (11)  (111)    (22)        (11111)
           (1(11))  (1111)      ((11)3)
                    (2(11))     (1(22))
                    (1(111))    (2(111))
                    (11(11))    (1(1111))
                    ((11)(11))  (11(111))
                    (1(1(11)))  (111(11))
                                (1(2(11)))
                                (2(1(11)))
                                (1(1(111)))
                                (1(11)(11))
                                (1(11(11)))
                                (11(1(11)))
                                (1((11)(11)))
                                (1(1(1(11))))

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.

The non-locally disjoint version is A141268.
Locally disjoint trees counted by vertices are A316473.
The case where all leaves are 1's is A316697.
Number of trees counted by A331678 with all atoms equal to 1.
Matula-Goebel numbers of locally disjoint rooted trees are A316495.
Unlabeled lone-child-avoiding locally disjoint rooted trees are A331680.
Cf. A000081, A000669, A001678, A005804, A060356, A300660, A316471, A316694, A316696, A319312, A330465, A331681.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
usot[n_]:=Prepend[Join@@Table[Select[Union[Sort/@Tuples[usot/@ptn]],disjointQ[DeleteCases[#,_?AtomQ]]&&SameQ@@Select[#,AtomQ]&],{ptn,Select[IntegerPartitions[n],Length[#]>1&]}],n];
Table[Length[usot[n]],{n,12}]

A331680 Number of lone-child-avoiding locally disjoint unlabeled rooted trees with n vertices.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 6, 9, 16, 26, 45, 72, 124, 201, 341, 561, 947, 1571, 2651, 4434, 7496, 12631, 21423, 36332, 61910, 105641, 180924, 310548, 534713, 923047

Offset: 1

Gus Wiseman, Jan 25 2020

First differs from A320268 at a(11) = 45, A320268(11) = 44.

A rooted tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other unequal child of the same vertex. Lone-child-avoiding means there are no unary branchings.

			The a(1) = 1 through a(9) = 16 trees (empty column indicated by dot):
  o  .  (oo)  (ooo)  (oooo)   (ooooo)   (oooooo)    (ooooooo)    (oooooooo)
                     (o(oo))  (o(ooo))  (o(oooo))   (o(ooooo))   (o(oooooo))
                              (oo(oo))  (oo(ooo))   (oo(oooo))   (oo(ooooo))
                                        (ooo(oo))   (ooo(ooo))   (ooo(oooo))
                                        ((oo)(oo))  (oooo(oo))   (oooo(ooo))
                                        (o(o(oo)))  (o(o(ooo)))  (ooooo(oo))
                                                    (o(oo)(oo))  ((ooo)(ooo))
                                                    (o(oo(oo)))  (o(o(oooo)))
                                                    (oo(o(oo)))  (o(oo(ooo)))
                                                                 (o(ooo(oo)))
                                                                 (oo(o(ooo)))
                                                                 (oo(oo)(oo))
                                                                 (oo(oo(oo)))
                                                                 (ooo(o(oo)))
                                                                 (o((oo)(oo)))
                                                                 (o(o(o(oo))))

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.

The enriched version is A316696.
The Matula-Goebel numbers of these trees are A331871.
The non-locally disjoint version is A001678.
These trees counted by number of leaves are A316697.
The semi-lone-child-avoiding version is A331872.
Cf. A000081, A000669, A005804, A060356, A141268, A300660, A316471, A316473, A316694, A316495, A319312, A330465, A331679, A331681, A331683.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
strut[n_]:=If[n==1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[strut/@c]]]/@Rest[IntegerPartitions[n-1]],disjointQ]];
Table[Length[strut[n]],{n,10}]

cp's OEIS Frontend

A339645 Triangle read by rows: T(n,k) is the number of inequivalent colorings of lone-child-avoiding rooted trees with n colored leaves using exactly k colors.

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A300660 Number of unlabeled rooted phylogenetic trees with n (leaf-) nodes such that for each inner node all children are either leaves or roots of distinct subtrees.

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A320160 Number of series-reduced balanced rooted trees whose leaves form an integer partition of n.

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A330465 Number of non-isomorphic series-reduced rooted trees whose leaves are multisets with a total of n elements.

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A320154 Number of series-reduced balanced rooted trees whose leaves form a set partition of {1,...,n}.

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A331686 Number of lone-child-avoiding locally disjoint rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.

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A316624 Number of balanced p-trees with n leaves.

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A320328 Number of square multiset partitions of integer partitions of n.

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