A358580 -id:A358580 - cp's OEIS frontend

A185650 a(n) is the number of rooted trees with 2n vertices n of whom are leaves.

1, 2, 8, 39, 214, 1268, 7949, 51901, 349703, 2415348, 17020341, 121939535, 885841162, 6511874216, 48359860685, 362343773669, 2736184763500, 20805175635077, 159174733727167, 1224557214545788, 9467861087020239, 73534456468877012, 573484090227222260

Offset: 1

Stepan Orevkov, Aug 29 2013

nonn

			From _Gus Wiseman_, Nov 27 2022: (Start)
The a(1) = 1 through a(3) = 8 rooted trees:
  (o)  ((oo))  (((ooo)))
       (o(o))  ((o)(oo))
               ((o(oo)))
               ((oo(o)))
               (o((oo)))
               (o(o)(o))
               (o(o(o)))
               (oo((o)))
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..200
V. M. Kharlamov and S. Yu. Orevkov, The number of trees half of whose vertices are leaves and asymptotic enumeration of plane real algebraic curves, arXiv:math/0301245 [math.AG], 2003; J. of Combinatorial Theory, Ser. A, 105 (2004), 127-142.
Index entries for sequences related to rooted trees

The ordered version is A000891, ranked by A358579.
This is the central column of A055277.
These trees are ranked by A358578.
For height = internals we have A358587.
Square trees are counted by A358589.
A000081 counts rooted trees, ordered A000108.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
A358575 counts rooted trees by nodes and internals, ordered A090181.
Cf. A034781, A109129, A358580, A358581-A358584, A358591.

terms = 23;
m = 2 terms;
T[, ] = 0;
Do[T[x_, z_] = z x - x + x Exp[Sum[Series[1/k T[x^k, z^k], {x, 0, j}, {z, 0, j}], {k, 1, j}]] // Normal, {j, 1, m}];
cc = CoefficientList[#, z]& /@ CoefficientList[T[x, z] , x];
Table[cc[[2n+1, n+1]], {n, 1, terms}] (* Jean-François Alcover, Sep 14 2018 *)
art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[art[n],Count[#,{},{-2}]==n/2&]],{n,2,10,2}] (* Gus Wiseman, Nov 27 2022 *)

\\ here R is A055277 as vector of polynomials
R(n) = {my(A = O(x)); for(j=1, n, A = x*(y - 1  + exp( sum(i=1, j, 1/i * subst( subst( A + x * O(x^(j\i)), x, x^i), y, y^i) ) ))); Vec(A)};
{my(A=R(2*30)); vector(#A\2, k, polcoeff(A[2*k],k))} \\ Andrew Howroyd, May 21 2018

Terms a(20) and beyond from Andrew Howroyd, May 21 2018

A358577 Matula-Goebel numbers of "square" rooted trees, i.e., whose height equals their number of leaves.

Original entry on oeis.org

1, 4, 12, 14, 18, 19, 21, 27, 40, 52, 60, 68, 70, 74, 78, 86, 89, 90, 91, 92, 95, 100, 102, 105, 107, 111, 117, 119, 122, 129, 130, 134, 135, 138, 146, 150, 151, 153, 161, 163, 169, 170, 175, 176, 181, 183, 185, 195, 201, 206, 207, 215, 219, 221, 225, 227, 230

Offset: 1

Gus Wiseman, Nov 25 2022

nonn

The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

			The terms together with their corresponding rooted trees begin:
   1: o
   4: (oo)
  12: (oo(o))
  14: (o(oo))
  18: (o(o)(o))
  19: ((ooo))
  21: ((o)(oo))
  27: ((o)(o)(o))
  40: (ooo((o)))
  52: (oo(o(o)))
  60: (oo(o)((o)))
  68: (oo((oo)))
  70: (o((o))(oo))
  74: (o(oo(o)))
  78: (o(o)(o(o)))
  86: (o(o(oo)))
  89: ((ooo(o)))
  90: (o(o)(o)((o)))

Gus Wiseman, The first 64 square rooted trees.

Internals instead of leaves: A358576, counted by A358587, ordered A358588.
Internals instead of height: A358578, counted by A185650, ordered A358579.
These trees are counted by A358589, ordered A358590.
A000081 counts rooted trees, ordered A000108.
A034781 counts trees by nodes and height.
A055277 counts trees by nodes and leaves, ordered A001263.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
Cf. A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A206487, A358379, A358580, A358581-A358586, A358592.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Count[MGTree[#],{},{0,Infinity}]==Depth[MGTree[#]]-1&]

A358552(a(n)) = A109129(a(n)).

A358578 Matula-Goebel numbers of rooted trees whose number of leaves equals their number of internal (non-leaf) nodes.

Original entry on oeis.org

2, 6, 7, 18, 20, 21, 26, 34, 37, 43, 54, 60, 63, 67, 70, 78, 88, 91, 92, 95, 102, 111, 116, 119, 122, 129, 142, 146, 151, 162, 164, 173, 180, 181, 189, 200, 201, 202, 210, 227, 234, 236, 239, 245, 260, 264, 269, 273, 276, 278, 285, 306, 308, 314, 322, 333, 337

Offset: 1

Gus Wiseman, Nov 25 2022

nonn

The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

			The terms together with their corresponding rooted trees begin:
   2: (o)
   6: (o(o))
   7: ((oo))
  18: (o(o)(o))
  20: (oo((o)))
  21: ((o)(oo))
  26: (o(o(o)))
  34: (o((oo)))
  37: ((oo(o)))
  43: ((o(oo)))
  54: (o(o)(o)(o))
  60: (oo(o)((o)))
  63: ((o)(o)(oo))
  67: (((ooo)))
  70: (o((o))(oo))
  78: (o(o)(o(o)))
  88: (ooo(((o))))
  91: ((oo)(o(o)))

Gus Wiseman, The first 64 rooted trees whose number of leaves equals their number of internal nodes.

These trees are counted by A185650, ordered A358579.
Height instead of leaves: A358576, counted by A358587, ordered A358588.
Height instead of internals: A358577, counted by A358589, ordered A358590.
Positions of 0's in A358580.
A000081 counts rooted trees, ordered A000108.
A034781 counts trees by nodes and height.
A055277 counts trees by nodes and leaves, ordered A001263.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
Cf. A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A206487, A358371, A358581-A358586, A358592.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Count[MGTree[#],{},{0,Infinity}]==Count[MGTree[#],[_],{0,Infinity}]&]

A342507(a(n)) = A109129(a(n)).

A358576 Matula-Goebel numbers of rooted trees whose node-height equals their number of internal (non-leaf) nodes.

Original entry on oeis.org

9, 15, 18, 21, 23, 30, 33, 35, 36, 39, 42, 46, 47, 49, 51, 57, 60, 61, 66, 70, 72, 73, 77, 78, 83, 84, 87, 91, 92, 93, 94, 95, 98, 102, 111, 113, 114, 119, 120, 122, 123, 129, 132, 133, 137, 140, 144, 146, 149, 151, 154, 156, 159, 166, 167, 168, 174, 177, 181

Offset: 1

Gus Wiseman, Nov 25 2022

nonn

The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

Node-height is the number of nodes in the longest path from root to leaf.

			The terms together with their corresponding rooted trees begin:
   9: ((o)(o))
  15: ((o)((o)))
  18: (o(o)(o))
  21: ((o)(oo))
  23: (((o)(o)))
  30: (o(o)((o)))
  33: ((o)(((o))))
  35: (((o))(oo))
  36: (oo(o)(o))
  39: ((o)(o(o)))
  42: (o(o)(oo))
  46: (o((o)(o)))
  47: (((o)((o))))
  49: ((oo)(oo))
  51: ((o)((oo)))
  57: ((o)(ooo))
  60: (oo(o)((o)))
  61: ((o(o)(o)))

Gus Wiseman, The first 64 rooted trees whose node-height equals their number of internal nodes.

The version for edge-height is A209638.
Square trees are A358577, counted by A358589, ordered A358590.
The version for leaves instead of height is A358578, counted by A185650.
These trees are counted by A358587, ordered A358588.
A000081 counts rooted trees, ordered A000108.
A034781 counts rooted trees by nodes and height.
A055277 counts rooted trees by leaves, ordered A001263.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
Cf. A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A206487, A358580, A358581-A358586, A358592, A358729.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Count[MGTree[#],[_],{0,Infinity}]==Depth[MGTree[#]]-1&]

A358552(a(n)) = A342507(a(n)).

A358586 Number of ordered rooted trees with n nodes, at least half of which are leaves.

Original entry on oeis.org

1, 1, 1, 4, 7, 31, 66, 302, 715, 3313, 8398, 39095, 104006, 484706, 1337220, 6227730, 17678835, 82204045, 238819350, 1108202513, 3282060210, 15195242478, 45741281820, 211271435479, 644952073662, 2971835602526, 9183676536076, 42217430993002, 131873975875180, 604834233372884

Offset: 1

Gus Wiseman, Nov 24 2022

nonn

			The a(1) = 1 through a(5) = 7 ordered trees:
  o  (o)  (oo)  (ooo)   (oooo)
                ((o)o)  ((o)oo)
                ((oo))  ((oo)o)
                (o(o))  ((ooo))
                        (o(o)o)
                        (o(oo))
                        (oo(o))

Andrew Howroyd, Table of n, a(n) for n = 1..1000

For equality we have A000891, unordered A185650.
Odd-indexed terms appear to be A065097.
The unordered version is A358583.
The opposite is the same, unordered A358584.
The strict case is A358585, unordered A358581.
A000108 counts ordered rooted trees, unordered A000081.
A001263 counts ordered rooted trees by nodes and leaves, unordered A055277.
A080936 counts ordered rooted trees by nodes and height, unordered A034781.
A090181 counts ordered rooted trees by nodes and internals, unord. A358575.
A358590 counts square ordered trees, unordered A358589 (ranked by A358577).
Cf. A109129, A342507, A358371, A358579, A358580, A358582, A358584, A358588.

aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n],Count[#,{},{0,Infinity}]>=Count[#,[_],{0,Infinity}]&]],{n,1,10}]

a(n) = if(n==1, 1, n--; (binomial(2*n,n)/(n+1) + if(n%2, binomial(n, (n-1)/2)^2 / n))/2) \\ Andrew Howroyd, Jan 13 2024

From Andrew Howroyd, Jan 13 2024: (Start)
a(n) = Sum_{k=1..floor(n/2)} A001263(n-1, k) for n >= 2.
a(2*n) = (A000108(2*n-1) + A000891(n-1))/2 for n >= 1;
a(2*n+1) = A000108(2*n)/2 for n >= 1. (End)

a(16) onwards from Andrew Howroyd, Jan 13 2024

A358581 Number of rooted trees with n nodes, most of which are leaves.

Original entry on oeis.org

1, 0, 1, 1, 4, 5, 20, 28, 110, 169, 663, 1078, 4217, 7169, 27979, 49191, 191440, 345771, 1341974, 2477719, 9589567, 18034670, 69612556, 132984290, 511987473, 991391707, 3807503552, 7460270591, 28585315026, 56595367747, 216381073935, 432396092153

Offset: 1

Gus Wiseman, Nov 23 2022

nonn

			The a(1) = 1 through a(7) = 20 trees:
  o  .  (oo)  (ooo)  (oooo)   (ooooo)   (oooooo)
                     ((ooo))  ((oooo))  ((ooooo))
                     (o(oo))  (o(ooo))  (o(oooo))
                     (oo(o))  (oo(oo))  (oo(ooo))
                              (ooo(o))  (ooo(oo))
                                        (oooo(o))
                                        (((oooo)))
                                        ((o)(ooo))
                                        ((o(ooo)))
                                        ((oo)(oo))
                                        ((oo(oo)))
                                        ((ooo(o)))
                                        (o((ooo)))
                                        (o(o)(oo))
                                        (o(o(oo)))
                                        (o(oo(o)))
                                        (oo((oo)))
                                        (oo(o)(o))
                                        (oo(o(o)))
                                        (ooo((o)))

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

For equality we have A185650 aerated, ranked by A358578.
The opposite version is A358582, non-strict A358584.
The non-strict version is A358583.
The ordered version is A358585, odd-indexed terms A065097.
A000081 counts rooted trees, ordered A000108.
A034781 counts rooted trees by nodes and height, ordered A080936.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
A358575 counts rooted trees by nodes and internal nodes, ordered A090181.
A358589 counts square trees, ranked by A358577, ordered A358590.
Cf. A000891, A109129, A342507, A358579, A358580, A358586, A358591.

art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[art[n],Count[#,{},{0,Infinity}]>Count[#,[_],{0,Infinity}]&]],{n,0,10}]

\\ See A358584 for R(n).
seq(n) = {my(A=R(n)); vector(n, n, my(u=Vecrev(A[n]/y)); vecsum(u[n\2+1..#u]))} \\ Andrew Howroyd, Dec 31 2022

A358581(n) + A358584(n) = A000081(n).
A358582(n) + A358583(n) = A000081(n).
a(n) = Sum_{k=floor(n/2)+1..n} A055277(n, k). - Andrew Howroyd, Dec 31 2022

Terms a(19) and beyond from Andrew Howroyd, Dec 31 2022

A358579 Numbers k such that the k-th standard ordered rooted tree has the same number of leaves as internal (non-leaf) nodes.

Original entry on oeis.org

2, 6, 7, 9, 20, 22, 23, 26, 27, 29, 35, 41, 66, 76, 78, 79, 84, 86, 87, 90, 91, 93, 97, 102, 103, 106, 107, 109, 115, 117, 130, 136, 138, 139, 141, 146, 153, 163, 169, 193, 196, 197, 201, 226, 241, 262, 263, 296, 300, 302, 303, 308, 310, 311, 314, 315, 317

Offset: 1

Gus Wiseman, Nov 25 2022

nonn

We define the n-th standard ordered rooted tree to be obtained by taking the (n-1)-th composition in standard order (graded reverse-lexicographic, A066099) as root and replacing each part with its own standard ordered rooted tree. This ranking is an ordered variation of Matula-Goebel numbers, giving a bijective correspondence between positive integers and unlabeled ordered rooted trees.

			The terms together with their corresponding rooted trees begin:
   2: (o)
   6: (o(o))
   7: ((oo))
   9: ((o)(o))
  20: (oo((o)))
  22: (o(((o))))
  23: (((o)(o)))
  26: (o(o(o)))
  27: ((o)(o)(o))
  29: ((o((o))))
  35: (((o))(oo))
  41: (((o(o))))
  66: (o(o)(((o))))
  76: (oo(ooo))
  78: (o(o)(o(o)))
  79: ((o(((o)))))
  84: (oo(o)(oo))
  86: (o(o(oo)))

These ordered trees are counted by A000891.
The unordered version is A358578, counted by A185650.
Height instead of leaves: counted by A358588, unordered A358576.
Height instead of internals: counted by A358590, unordered A358577.
Standard ordered tree number statistics: A358371, A358372, A358379, A358553.
A000081 counts rooted trees, ordered A000108.
A055277 counts trees by nodes and leaves, ordered A001263.
Cf. A014221, A206487, A358373, A358580, A358587, A358589.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
Select[Range[100],Count[srt[#],{},{0,Infinity}]==Count[srt[#],[_],{0,Infinity}]&]

A358371(a(n)) = A358553(a(n)).

A358592 Matula-Goebel numbers of rooted trees whose height, number of leaves, and number of internal (non-leaf) nodes are all equal.

Original entry on oeis.org

18, 21, 60, 70, 78, 91, 92, 95, 102, 111, 119, 122, 129, 146, 151, 181, 201, 227, 264, 269, 308, 348, 376, 406, 418, 426, 452, 492, 497, 519, 551, 562, 574, 583, 596, 606, 659, 664, 668, 698, 707, 708, 717, 779, 794, 796, 809, 826, 834, 911, 932, 934, 942, 958

Offset: 1

Gus Wiseman, Nov 25 2022

nonn

The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

			The terms together with their corresponding rooted trees begin:
   18: (o(o)(o))
   21: ((o)(oo))
   60: (oo(o)((o)))
   70: (o((o))(oo))
   78: (o(o)(o(o)))
   91: ((oo)(o(o)))
   92: (oo((o)(o)))
   95: (((o))(ooo))
  102: (o(o)((oo)))
  111: ((o)(oo(o)))
  119: ((oo)((oo)))
  122: (o(o(o)(o)))
  129: ((o)(o(oo)))
  146: (o((o)(oo)))
  151: ((oo(o)(o)))
  181: ((o(o)(oo)))
  201: ((o)((ooo)))
  227: (((oo)(oo)))

Gus Wiseman, The first 64 ordered trees whose height, number of leaves, and number of internal nodes are all equal.

Any number of leaves: A358576, counted by A358587 (ordered A358588).
Any number of internals: A358577, counted by A358589, ordered A358590.
Any height: A358578, ordered A358579, counted by A185650.
A000081 counts rooted trees, ordered A000108.
A034781 counts rooted trees by nodes and height.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
Cf. A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A206487, A358580, A358581-A358586.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Count[MGTree[#],[_],{0,Infinity}]==Count[MGTree[#],{},{0,Infinity}]==Depth[MGTree[#]]-1&]

A358552(a(n)) = A342507(a(n)) = A109129(a(n)).

A358729 Difference between the number of nodes and the node-height of the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 2, 2, 1, 0, 2, 1, 2, 2, 3, 1, 3, 2, 2, 3, 1, 2, 3, 3, 2, 4, 3, 1, 3, 0, 4, 2, 2, 3, 4, 2, 3, 3, 3, 1, 4, 2, 2, 4, 3, 2, 4, 4, 4, 3, 3, 3, 5, 3, 4, 4, 2, 1, 4, 3, 1, 5, 5, 4, 3, 2, 3, 4, 4, 2, 5, 3, 3, 5, 4, 3, 4, 1, 4, 6, 2, 2, 5, 4, 3, 3, 3, 3, 5, 4, 4, 2, 3, 4, 5, 3, 5, 4, 5, 2, 4, 4, 4, 5, 4, 3, 6

Offset: 1

Gus Wiseman, Nov 29 2022

nonn

Node-height is the number of nodes in the longest path from root to leaf.
The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.
Because the number of distinct terminal subtrees of the rooted tree with Matula-Goebel number n, i.e., A317713(n) (= 1+A324923(n)), is always at least one larger than the depth of the same tree (= A109082(n)), it follows that a(n) >= A366386(n) for all n. - Antti Karttunen, Oct 23 2023

			The tree (oo(oo(o))) with Matula-Goebel number 148 has 8 nodes and node-height 4, so a(148) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences related to Matula-Goebel numbers

Positions of 0's are A007097.
Positions of first appearances are A358730.
Positions of 1's are A358731.
Other differences: A358580, A358724, A358726.
A000081 counts rooted trees, ordered A000108.
A034781 counts rooted trees by nodes and height, ordered A080936.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
MG core: A000040, A000720, A001222, A056239, A112798.
Cf. A206487, A209638, A316321, A358576, A358577, A358592, A358725, A366386.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Count[MGTree[n],_,{0,Infinity}]-(Depth[MGTree[n]]-1),{n,100}]

A061775(n) = if(1==n, 1, if(isprime(n), 1+A061775(primepi(n)), {my(pfs,t,i); pfs=factor(n); pfs[,1]=apply(t->A061775(t),pfs[,1]); (1-bigomega(n)) + sum(i=1, omega(n), pfs[i,1]*pfs[i,2])}));
A358552(n) = { my(v=factor(n)[, 1], d=0); while(#v, d++; v=fold(setunion, apply(p->factor(primepi(p))[, 1]~, v))); (1+d); }; \\ (after program given in A109082 by Kevin Ryde, Sep 21 2020)
A358729(n) = (A061775(n)-A358552(n)); \\ Antti Karttunen, Oct 23 2023

a(n) = A061775(n) - A358552(n).

a(n) = A196050(n) - A109082(n). - Antti Karttunen, Oct 23 2023

Data section extended up to a(108) by Antti Karttunen, Oct 23 2023

A358584 Number of rooted trees with n nodes, at most half of which are leaves.

Original entry on oeis.org

0, 1, 1, 3, 5, 15, 28, 87, 176, 550, 1179, 3688, 8269, 25804, 59832, 186190, 443407, 1375388, 3346702, 10348509, 25632265, 79020511, 198670299, 610740694, 1555187172, 4768244803, 12276230777, 37546795678, 97601239282, 297831479850, 780790439063, 2377538260547

Offset: 1

Gus Wiseman, Nov 23 2022

nonn

			The a(2) = 1 through a(6) = 15 trees:
  (o)  ((o))  ((oo))   (((oo)))   (((ooo)))
              (o(o))   ((o)(o))   ((o)(oo))
              (((o)))  ((o(o)))   ((o(oo)))
                       (o((o)))   ((oo(o)))
                       ((((o))))  (o((oo)))
                                  (o(o)(o))
                                  (o(o(o)))
                                  (oo((o)))
                                  ((((oo))))
                                  (((o)(o)))
                                  (((o(o))))
                                  ((o)((o)))
                                  ((o((o))))
                                  (o(((o))))
                                  (((((o)))))

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

For equality we have A185650 aerated, ranked by A358578.
The complement is A358581.
The strict case is A358582.
The opposite version is A358583.
A000081 counts rooted trees, ordered A000108.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
A358575 counts rooted trees by nodes and internal nodes, ordered A090181.
A358589 counts square trees, ranked by A358577, ordered A358590.
Cf. A000891, A034781, A065097, A109129, A342507, A358579, A358580, A358585, A358586, A358591.

art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[art[n],Count[#,{},{0,Infinity}]<=Count[#,[_],{0,Infinity}]&]],{n,0,10}]

R(n) = {my(A = O(x)); for(j=1, n, A = x*(y - 1  + exp( sum(i=1, j, 1/i * subst( subst( A + O(x*x^(j\i)), x, x^i), y, y^i) ) ))); Vec(A)};
seq(n) = {my(A=R(n)); vector(n, n, vecsum(Vecrev(A[n]/y)[1..n\2]))} \\ Andrew Howroyd, Dec 30 2022

A358581(n) + A358584(n) = A000081(n).
A358582(n) + A358583(n) = A000081(n).
a(n) = Sum_{k=1..floor(n/2)} A055277(n, k). - Andrew Howroyd, Dec 30 2022

Terms a(19) and beyond from Andrew Howroyd, Dec 30 2022

cp's OEIS Frontend

A185650 a(n) is the number of rooted trees with 2n vertices n of whom are leaves.

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A358577 Matula-Goebel numbers of "square" rooted trees, i.e., whose height equals their number of leaves.

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A358578 Matula-Goebel numbers of rooted trees whose number of leaves equals their number of internal (non-leaf) nodes.

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A358576 Matula-Goebel numbers of rooted trees whose node-height equals their number of internal (non-leaf) nodes.

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A358586 Number of ordered rooted trees with n nodes, at least half of which are leaves.

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A358581 Number of rooted trees with n nodes, most of which are leaves.

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A358579 Numbers k such that the k-th standard ordered rooted tree has the same number of leaves as internal (non-leaf) nodes.

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A358592 Matula-Goebel numbers of rooted trees whose height, number of leaves, and number of internal (non-leaf) nodes are all equal.