A206487 -id:A206487 - cp's OEIS frontend

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

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

A214569 Irregular triangle read by rows: T(n,k) is the number of rooted trees having n vertices and isomorphic (as rooted trees) to k ordered trees (n>=1, k>=1).

Original entry on oeis.org

1, 1, 2, 3, 1, 5, 3, 1, 6, 8, 4, 2, 10, 17, 7, 8, 1, 5, 11, 34, 16, 25, 3, 18, 0, 3, 1, 1, 0, 3, 16, 63, 27, 65, 6, 56, 1, 16, 5, 4, 0, 22, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 1, 19, 111, 47, 154, 12, 138, 3, 65, 13, 13, 0, 95, 0, 0, 3, 5, 0, 13, 0, 8, 1, 0, 0, 13, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 26, 186, 73, 348, 18, 319, 6, 208, 35, 32, 0, 308, 0, 2, 13, 34, 0, 58, 0, 29, 1, 0, 0, 88, 0, 0, 1, 1, 0, 16, 0, 0, 0, 0, 1, 18, 0, 0, 0, 8, 0, 2, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6

Offset: 1

Emeric Deutsch, Jul 28 2012

Row n contains A214570(n) entries.
T(n,1) = A003238(n).
Sum(T(n,k), k=1..n) = A000081(n) = number of rooted trees with n vertices.
Sum(k*T(n,k), k=1..n) = A000108(n-1) (the Catalan numbers).
T(n,k) is also the number of size k equivalence classes of function representations as x^x^...^x with n x's and parentheses inserted in all possible ways. T(4,2) = 1: (x^x)^(x^x) == (x^(x^x))^x; T(5,3) = 1: ((x^x)^x)^(x^x) == ((x^x)^(x^x))^x == ((x^(x^x))^x)^x. - Alois P. Heinz, Aug 31 2012

			Row 4 is 3,1: among the four rooted trees with 4 vertices the path tree P_4, the star tree K_{1,3}, and the tree in the shape of Y are isomorphic only to themselves, while A - B - C - D with root at B is isomorphic to itself and to A - B - C - D with root at C.
Triangle starts:
   1;
   1;
   2;
   3,  1;
   5,  3,  1;
   6,  8,  4,  2;
  10, 17,  7,  8, 1,  5;
  11, 34, 16, 25, 3, 18, 0, 3, 1, 1, 0, 3;
  ...

Alois P. Heinz, Rows n = 1..16, flattened

Cf. A000108, A000081, A003238, A214570, A214571, A061775, A206487, A215703.

F:= proc(n) option remember; `if`(n=1, [x+1],
      [seq(seq(seq(f^g, g=F(n-i)), f=F(i)), i=1..n-1)])
    end:
T:= proc(n) option remember; local i, l, p;
      l:= map(f->coeff(series(f, x, n+1), x, n), F(n)):
      p:= proc() 0 end: forget(p);
      for i in l do p(i):= p(i)+1 od:
      l:= map(p, l); forget(p);
      for i in l do p(i):= p(i)+1 od:
      seq(p(i)/i, i=1..max(l[]))
    end:
seq(T(n), n=1..10);  # Alois P. Heinz, Aug 31 2012

F[n_] := F[n] = If[n == 1, {x+1}, Flatten[Table[Table[Table[f^g, {g, F[n-i]}], {f, F[i]}], {i, 1, n-1}]]]; T[n_] := T[n] = Module[{i, l, p}, l = Map[Function[ {f}, Coefficient[Series[f, {x, 0, n+1}], x, n]], F[n]]; Clear[p]; p[] = 0; Do[ p[i] = p[i]+1 , {i, l}]; l = Map[p, l]; Clear[p]; p[] = 0; Do[p[i] = p[i]+1, {i, l}]; Table[p[i]/i, {i, 1, Max[l]}]]; Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, May 28 2015, after Alois P. Heinz *)

No formula available. Entries have been obtained by counting (using Maple) the rooted trees (identified by their Matula-Goebel numbers) with the required properties (using A061775 and A206487).

A358506 Matula-Goebel number of the n-th standard ordered rooted tree.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 6, 8, 7, 10, 9, 12, 10, 12, 12, 16, 11, 14, 15, 20, 15, 18, 18, 24, 14, 20, 18, 24, 20, 24, 24, 32, 13, 22, 21, 28, 25, 30, 30, 40, 21, 30, 27, 36, 30, 36, 36, 48, 22, 28, 30, 40, 30, 36, 36, 48, 28, 40, 36, 48, 40, 48, 48, 64, 13, 26, 33, 44

Offset: 1

Gus Wiseman, Nov 20 2022

nonn

First differs from A333219 at a(65) = 13, A333219(65) = 17.
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.
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 first eight standard ordered trees are: o, (o), ((o)), (oo), (((o))), ((o)o), (o(o)), (ooo), with Matula-Goebel numbers: 1, 2, 3, 4, 5, 6, 6, 8.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

For binary instead of standard encoding we have A127301.
There are exactly A206487(n) appearances of n.
For binary instead of Matula-Goebel encoding we have A358505.
Positions of first appearances are A358522, sorted A358521.
A000108 counts ordered rooted trees, unordered A000081.
A214577 and A358377 rank trees with no permutations.
Cf. A001263, A014486, A061775, A196050, A358371, A358372, A358378, A358379, A358507, A358508.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
mgnum[t_]:=If[t=={},1,Times@@Prime/@mgnum/@t];
Table[mgnum[srt[n]],{n,100}]

A358724 Difference between the number of internal (non-leaf) nodes and the edge-height of the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 1, 1, 0, 1, 2, 1, 0, 0, 2, 2, 0, 1, 0, 0, 1, 1, 0, 2, 0, 2, 1, 0, 0, 2, 1, 0, 1, 1, 0, 3, 0, 1, 1, 0, 0, 3, 0, 1, 1, 2, 0, 1

Offset: 1

Gus Wiseman, Nov 29 2022

nonn

Edge-height (A109082) is the number of edges 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.

			The tree (o(o)((o))(oo)) with Matula-Goebel number 210 has edge-height 3 and 5 internal nodes, so a(210) = 2.

Positions of 0's are A209638, complement A358725.
Positions of 1's are A358576, counted by A358587.
Other differences: A358580, A358726, A358729.
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, A007097, A056239, A112798.
Cf. A185650, A206487, A316321, A358577, A358578, A358592, A358730.

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]]-2),{n,100}]

a(n) = A342507(n) - A109082(n).

A358726 Difference between the node-height and the number of leaves in the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

0, 1, 2, 0, 3, 1, 1, -1, 1, 2, 4, 0, 2, 0, 2, -2, 2, 0, 0, 1, 0, 3, 2, -1, 2, 1, 0, -1, 3, 1, 5, -3, 3, 1, 1, -1, 1, -1, 1, 0, 3, -1, 1, 2, 1, 1, 3, -2, -1, 1, 1, 0, -1, -1, 3, -2, -1, 2, 3, 0, 1, 4, -1, -4, 1, 2, 1, 0, 1, 0, 2, -2, 1, 0, 1, -2, 2, 0, 4, -1

Offset: 1

Gus Wiseman, Nov 29 2022

sign

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.

			The tree (oo(oo(o))) with Matula-Goebel number 148 has node-height 4 and 5 leaves, so a(148) = -1.

Positions of first appearances are A007097 and latter terms of A000079.
Positions of 0's are A358577.
Other differences: A358580, A358724, A358729.
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. A185650, A206487, A209638, A358576, A358578, A358587, A358730.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[(Depth[MGTree[n]]-1)-Count[MGTree[n],{},{0,Infinity}],{n,1000}]

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

A214571 Irregular triangle read by rows: T(n,k) is the number of ordered trees having n vertices and isomorphic (as rooted trees) to k ordered trees (n >= 1, k >= 1).

Original entry on oeis.org

1, 1, 2, 3, 2, 5, 6, 3, 6, 16, 12, 8, 10, 34, 21, 32, 5, 30, 11, 68, 48, 100, 15, 108, 0, 24, 9, 10, 0, 36, 16, 126, 81, 260, 30, 336, 7, 128, 45, 40, 0, 264, 0, 0, 15, 0, 0, 18, 0, 40, 0, 0, 0, 24, 19, 222, 141, 616, 60, 828, 21, 520, 117, 130, 0, 1140, 0, 0, 45, 80, 0, 234, 0, 160, 21, 0, 0, 312, 0, 0, 0, 0, 0, 120, 0, 0, 0, 0, 0, 36, 0, 0, 0, 40, 26, 372, 219, 1392, 90, 1914, 42, 1664, 315, 320, 0, 3696, 0, 28, 195, 544, 0, 1044, 0, 580, 21, 0, 0, 2112, 0, 0, 27, 28, 0, 480, 0, 0, 0, 0, 35, 648, 0, 0, 0, 320, 0, 84, 0, 0, 0, 0, 0, 240, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 360

Offset: 1

Emeric Deutsch, Jul 28 2012

Row n contains A214570(n) entries.
T(n,1) = A003238(n).
Sum_{k=1..n} T(n,k) = A000108(n) (the Catalan numbers).
Sum_{k=1..n} T(n,k)/k = A000081(n) (the number of rooted trees with n vertices).
T(n,k) = k*A214569(n,k).
T(n,k) is also the number of function representations as x^x^...^x with n x's and parentheses inserted in all possible ways that are equivalent to (describe the same function as) k-1 other representations. T(4,2) = 2: (x^x)^(x^x), (x^(x^x))^x; T(5,3) = 3: ((x^x)^x)^(x^x), ((x^x)^(x^x))^x, ((x^(x^x))^x)^x. - Alois P. Heinz, Aug 31 2012

			Row 4 is 3,2: among the five ordered trees with 4 vertices the path tree P_4, the star tree K_{1,3}, and the tree in the shape of Y are isomorphic only to themselves, while A - B - C - D with root at B and A - B - C - D with root at C are isomorphic among themselves.
Triangle starts:
   1;
   1;
   2;
   3,   2;
   5,   6,   3;
   6,  16,  12,   8;
  10,  34,  21,  32,   5,  30;
  11,  68,  48, 100,  15, 108,   0,  24,   9,  10,   0,  36;

Alois P. Heinz, Rows n = 1..16, flattened

Cf. A000108, A000081, A003238, A214569, A214570, A061775, A206487, A215703.

F:= proc(n) option remember; `if`(n=1, [x+1],
      [seq(seq(seq(f^g, g=F(n-i)), f=F(i)), i=1..n-1)])
    end:
T:= proc(n) option remember; local i, l, p;
      l:= map(f->coeff(series(f, x, n+1), x, n), F(n)):
      p:= proc() 0 end: forget(p);
      for i in l do p(i):= p(i)+1 od:
      l:= map(p, l); forget(p);
      for i in l do p(i):= p(i)+1 od:
      seq(p(i), i=1..max(l[]))
    end:
seq(T(n), n=1..10);  # Alois P. Heinz, Aug 31 2012

F[n_] := F[n] = If[n == 1, {x+1}, Flatten[Table[Table[Table[f^g, {g, F[n-i]}], {f, F[i]}], {i, 1, n-1}]]]; T[n_] := T[n] = Module[{i, l, p}, l = Map[Function[ {f}, Coefficient[Series[f, {x, 0, n+1}], x, n]], F[n]]; Clear[p]; p[] = 0; Do[ p[i] = p[i]+1 , {i, l}]; l = Map[p, l]; Clear[p]; p[] = 0; Do[p[i] = p[i]+1, {i, l}]; Table[p[i], {i, 1, Max[l]}]]; Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, May 28 2015, after Alois P. Heinz *)

No formula available. Entries have been obtained from T(n,k)= k*A214569(n,k).

A304938 a(n) is the smallest number which can be written in n different ways as an ordered product of prime factors.

Original entry on oeis.org

1, 6, 12, 24, 48, 30, 192, 384, 768, 72, 3072, 60, 12288, 24576, 144, 98304, 196608, 393216, 786432, 120, 288, 6291456, 12582912, 210, 50331648, 100663296, 201326592, 576, 805306368, 180, 3221225472, 6442450944, 12884901888, 25769803776, 432, 1152, 206158430208, 412316860416, 824633720832, 1649267441664

Offset: 1

Vincent Champain, May 21 2018

nonn

It can be easily demonstrated that a(n) exists for all n and is less than or equal to 2^(n-1)*3 since 2^(n-1)*3 can be written in n different ways.
If n is a prime number then a(n) = 2^(n-1)*3, but there are also nonprime numbers n with this property (e.g., 9 and 14).
If n=k! then a(n) is the product of the first k prime numbers.
Finding the terms up to 2^64 was the focus of the 4th question of the ICPC coding challenge in 2013.

			a(1) = 1 because only a prime power or the empty product (which equals 1) can be written in just one way, and no prime power is smaller than 1.
a(2) = 6 = 3 * 2 = 2 * 3 because none of 3, 4, 5 can be written in two different ways.
a(3) = 12 = 3 * 2 * 2 = 2 * 3 * 2 = 2 * 2 * 3 (each of 7, 8, 9, 10, 11 can be written in at most 2 ways).
a(4) = 24 = 2 * 2 * 2 * 3 (each of 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23 can be written in at most 3 ways).

Amiram Eldar, Table of n, a(n) for n = 1..136
The ACM-ICPC International Collegiate Programming Contest, ICPC 2013 problems

Cf. A000961, A007283 (2^n*3), A008480 (number of ordered prime factorizations of n).
Subsequence of A025487.
The sorted version is A358526.
Cf. A001222, A027746, A206487, A344606, A358508.

uv=Table[Length[Permutations[Join@@ConstantArray@@@FactorInteger[n]]],{n,1,1000}];
Table[Position[uv,k][[1,1]],{k,Min@@Complement[Range[Max@@uv],uv]-1}] (* Gus Wiseman, Nov 22 2022 *)

a008480(n) = my(f=factor(n)); sum(k=1, #f~, f[k,2])!/prod(k=1, #f~, f[k,2]!);
a(n) = {my(k=2); while (a008480(k) !=n, k++); k;} \\ Michel Marcus, May 23 2018

a(p) = 2^(p-1)*3 if p is a prime.
a(k!) = prime(k)# is the k-th primorial number. So for no m < k!, prime(k) | a(m). - David A. Corneth, May 24 2018
a(n) = min { k : A008480(k) = n }. - Alois P. Heinz, May 26 2018

A358508 Least Matula-Goebel number of a tree with exactly n permutations.

Original entry on oeis.org

1, 6, 12, 24, 48, 30, 192, 104, 148, 72, 3072, 60, 12288, 832, 144, 712, 196608, 222, 786432, 120, 288, 13312

Offset: 1

Gus Wiseman, Nov 20 2022

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.

To get a permutation of a tree, we choose a permutation of the multiset of branches of each node.

			The terms together with their corresponding trees begin:
      1: o
      6: (o(o))
     12: (oo(o))
     24: (ooo(o))
     48: (oooo(o))
     30: (o(o)((o)))
    192: (oooooo(o))
    104: (ooo(o(o)))
    148: (oo(oo(o)))
     72: (ooo(o)(o))
   3072: (oooooooooo(o))
     60: (oo(o)((o)))
  12288: (oooooooooooo(o))
    832: (oooooo(o(o)))
    144: (oooo(o)(o))
    712: (ooo(ooo(o)))

Position of first appearance of n in A206487.
The sorted version is A358507.
A000081 counts rooted trees, ordered A000108.
A214577 and A358377 rank trees with no permutations.
Cf. A001263, A032027, A061775, A127301, A196050, A358378, A358506, A358521, A358522.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]
MGTree[n_Integer]:=If[n===1,{},MGTree/@primeMS[n]]
treeperms[t_]:=Times @@ Cases[t,b:{}:>Length[Permutations[b]],{0,Infinity}];
uv=Table[treeperms[MGTree[n]],{n,100000}];
Table[Position[uv,k][[1,1]],{k,Min@@Complement[Range[Max@@uv],uv]-1}]

A358725 Matula-Goebel numbers of rooted trees with a greater number of internal (non-leaf) vertices than edge-height.

Original entry on oeis.org

9, 15, 18, 21, 23, 25, 27, 30, 33, 35, 36, 39, 42, 45, 46, 47, 49, 50, 51, 54, 55, 57, 60, 61, 63, 65, 66, 69, 70, 72, 73, 75, 77, 78, 81, 83, 84, 85, 87, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 102, 103, 105, 108, 110, 111, 113, 114, 115, 117, 119, 120, 121

Offset: 1

Gus Wiseman, Nov 29 2022

nonn

Edge-height (A109082) is the number of edges 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.

			The terms together with their corresponding trees begin:
   9: ((o)(o))
  15: ((o)((o)))
  18: (o(o)(o))
  21: ((o)(oo))
  23: (((o)(o)))
  25: (((o))((o)))
  27: ((o)(o)(o))
  30: (o(o)((o)))
  33: ((o)(((o))))
  35: (((o))(oo))
  36: (oo(o)(o))
  39: ((o)(o(o)))
  42: (o(o)(oo))
  45: ((o)(o)((o)))
  46: (o((o)(o)))
  47: (((o)((o))))
  49: ((oo)(oo))
  50: (o((o))((o)))

Gus Wiseman, The first 64 ordered trees with a greater number of internal vertices than edge-height.

Complement of A209638 (the case of equality).
These trees are counted by A316321.
Positions of positive terms in A358724.
The case of equality for node-height is A358576.
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.
Differences: A358580, A358724, A358726, A358729.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
MG core: A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A185650, A206487, A358577, A358578, A358581-A358586, A358587, A358592, A358730.

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[#]]-2&]

A342507(a(n)) > A109082(a(n)).

cp's OEIS Frontend

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

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

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A214569 Irregular triangle read by rows: T(n,k) is the number of rooted trees having n vertices and isomorphic (as rooted trees) to k ordered trees (n>=1, k>=1).

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A358506 Matula-Goebel number of the n-th standard ordered rooted tree.

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A358724 Difference between the number of internal (non-leaf) nodes and the edge-height of the rooted tree with Matula-Goebel number n.

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A358726 Difference between the node-height and the number of leaves in the rooted tree with Matula-Goebel number n.

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A214571 Irregular triangle read by rows: T(n,k) is the number of ordered trees having n vertices and isomorphic (as rooted trees) to k ordered trees (n >= 1, k >= 1).

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A304938 a(n) is the smallest number which can be written in n different ways as an ordered product of prime factors.