A358576 -id:A358576 - cp's OEIS frontend

A109082 Depth of rooted tree having Matula-Goebel number n.

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

Offset: 1

Keith Briggs, Aug 17 2005

nonn

Another term for depth is height.

Starting with n, a(n) is the number of times one must take the product of prime indices (A003963) to reach 1. - Gus Wiseman, Mar 27 2019

			a(7) = 2 because the rooted tree with Matula-Goebel number 7 is the 3-edge rooted tree Y of height 2.

François Marques, Table of n, a(n) for n = 1..10000 (terms 1..5381 from Alois P. Heinz).
Emeric Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to Matula-Goebel numbers

A left inverse of A007097.
Cf. A003963, A061775, A091233.
Cf. A000081, A000720, A001222, A109129, A112798, A196050, A290822, A317713, A320325, A324927 (positions of 2), A324928 (positions of 3), A325032.
This statistic is counted by A034781, ordered A080936.
The ordered version is A358379.
For node-height instead of edge-height we have A358552.
Cf. A000108, A055277, A056239, A342507, A358576.

with(numtheory): a := proc(n) option remember; if n = 1 then 0 elif isprime(n) then 1+a(pi(n)) else max((map (p->a(p), factorset(n)))[]) end if end proc: seq(a(n), n = 1 .. 100); # Emeric Deutsch, Sep 16 2011

a [n_] := a[n] = If[n == 1, 0, If[PrimeQ[n], 1+a[PrimePi[n]], Max[Map[a, FactorInteger[n][[All, 1]]]]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, May 06 2014, after Emeric Deutsch *)

a(n) = my(v=factor(n)[,1],d=0); while(#v,d++; v=fold(setunion, apply(p->factor(primepi(p))[,1]~, v))); d; \\ Kevin Ryde, Sep 21 2020

from functools import lru_cache
from sympy import isprime, primepi, primefactors
@lru_cache(maxsize=None)
def A109082(n):
    if n == 1 : return 0
    if isprime(n): return 1+A109082(primepi(n))
    return max(A109082(p) for p in primefactors(n)) # Chai Wah Wu, Mar 19 2022

a(1)=0; if n is the t-th prime, then a(n) = 1 + a(t); if n is composite, n=t*s, then a(n) = max(a(t),a(s)). The Maple program is based on this.
a(A007097(n)) = n.
a(n) = A358552(n) - 1. - Gus Wiseman, Nov 27 2022

Edited by Emeric Deutsch, Sep 16 2011

A342507 Number of internal nodes in rooted tree with Matula-Goebel number n.

Original entry on oeis.org

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

Offset: 1

François Marques, Mar 14 2021

nonn

The label f(T) for a rooted tree T is 1 if T has 1 node, otherwise f(T) = Product_{T_i} prime(f(T_i)) where the T_i are the subtrees obtained by deleting the root and the edges adjacent to it. (Cf. A061773 for illustration.)

			a(7) = 2 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y.
a(2^m) = 1 because the rooted tree with Matula-Goebel number 2^m is the star tree with m edges.

François Marques, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Matula-Goebel numbers

Other statistics are: A061775 (nodes), A109082 (edge-height), A109129 (leaves), A196050 (edges), A358552 (node-height).
An ordered version is A358553.
Positions of first appearances are A358554.
A000081 counts rooted trees, ordered A000108.
A358575 counts rooted trees by nodes and internals.
Cf. A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A034781, A055277, A206487, A358576, A358578, A358592.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Count[MGTree[n],[_],{0,Infinity}],{n,100}] (* Gus Wiseman, Nov 28 2022 *)

A342507(n) = if( n==1, 0, my(f=factor(n)); 1+sum(k=1,matsize(f)[1],A342507(primepi(f[k,1]))*f[k,2]));

a(1)=0 and a(n) = A061775(n) - A109129(n) for n > 1.

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)).

A358552 Node-height of the rooted tree with Matula-Goebel number n. Number of nodes in the longest path from root to leaf.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 26 2022

nonn

Edge-height is given by A109082 (see formula).

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 Matula-Goebel number of ((ooo(o))) is 89, and it has node-height 4, so a(89) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Positions of first appearances are A007097.
This statistic is counted by A034781, ordered A080936.
The ordered version is A358379(n) + 1.
A000081 counts rooted trees, ordered A000108.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
Other statistics: A061775 (nodes), A109082 (edge-height), A109129 (leaves), A196050 (edges), A342507 (internals).
Cf. A000040, A000720, A001222, A056239, A112798.
Cf. A206487, A358576, A358589, A358592, A358729.

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

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 Kevin Ryde in A109082) - Antti Karttunen, Oct 23 2023

from functools import lru_cache
from sympy import isprime, primepi, primefactors
@lru_cache(maxsize=None)
def A358552(n):
    if n == 1 : return 1
    if isprime(n): return 1+A358552(primepi(n))
    return max(A358552(p) for p in primefactors(n)) # Chai Wah Wu, Apr 15 2024

a(n) = A109082(n) + 1.

a(n) = A061775(n) - A358729(n). - Antti Karttunen, Oct 23 2023

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

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)).

A358589 Number of square rooted trees with n nodes.

Original entry on oeis.org

1, 0, 1, 0, 3, 2, 11, 17, 55, 107, 317, 720, 1938, 4803, 12707, 32311, 85168, 220879, 581112, 1522095, 4014186, 10568936, 27934075, 73826753, 195497427, 517927859, 1373858931, 3646158317, 9684878325, 25737819213, 68439951884, 182070121870, 484583900955, 1290213371950

Offset: 1

Gus Wiseman, Nov 23 2022

nonn

We say that a tree is square if it has the same height as number of leaves.

			The a(1) = 1 through a(7) = 11 trees:
  o  .  (oo)  .  ((ooo))  ((o)(oo))  (((oooo)))
                 (o(oo))  (o(o)(o))  ((o(ooo)))
                 (oo(o))             ((oo(oo)))
                                     ((ooo(o)))
                                     (o((ooo)))
                                     (o(o(oo)))
                                     (o(oo(o)))
                                     (oo((oo)))
                                     (oo(o(o)))
                                     (ooo((o)))
                                     ((o)(o)(o))

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

For internals instead of height we have A185650 aerated, ranked by A358578.
These trees are ranked by A358577.
For internals instead of leaves we have A358587, ranked by A358576.
The ordered version is A358590.
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.
Cf. A000891, A065097, A109129, A358552, A358588, 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}]==Depth[#]-1&]],{n,1,10}]

\\ R(n,f) enumerates trees by height(h), nodes(x) and leaves(y).
R(n,f) = {my(A=O(x*x^n), Z=0); for(h=1, n, my(p = A); A = x*(y - 1  + exp( sum(i=1, n-1, 1/i * subst( subst( A + O(x*x^((n-1)\i)), x, x^i), y, y^i) ) )); Z += f(h, A-p)); Z}
seq(n) = {Vec(R(n, (h,p)->polcoef(p,h,y)), -n)} \\ Andrew Howroyd, Jan 01 2023

Terms a(19) and beyond from Andrew Howroyd, Jan 01 2023

A358580 Difference between the number of leaves and the number of internal (non-leaf) nodes in the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 25 2022

sign

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 Matula-Goebel number of ((ooo(o))) is 89, and it has 4 leaves and 3 internal nodes, so a(89) = 1.

Zeros are A358578, counted by A185650 (ordered A358579).
Positions of positive terms are counted by A358581, negative A358582.
Positions of nonnegative terms are counted by A358583, nonpositive A358584.
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, A358576, A358577, A358592.

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

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

A358587 Number of n-node rooted trees of height equal to the number of internal (non-leaf) nodes.

Original entry on oeis.org

0, 0, 0, 0, 1, 4, 14, 41, 111, 282, 688, 1627, 3761, 8540, 19122, 42333, 92851, 202078, 436916, 939359, 2009781, 4281696, 9087670, 19223905, 40544951, 85284194, 178956984, 374691171, 782936761, 1632982372, 3400182458, 7068800357, 14674471611, 30422685030

Offset: 1

Gus Wiseman, Nov 23 2022

nonn

			The a(5) = 1 through a(7) = 14 trees:
  ((o)(o))  ((o)(oo))   ((o)(ooo))
            (o(o)(o))   ((oo)(oo))
            (((o)(o)))  (o(o)(oo))
            ((o)((o)))  (oo(o)(o))
                        (((o))(oo))
                        (((o)(oo)))
                        ((o)((oo)))
                        ((o)(o(o)))
                        ((o(o)(o)))
                        (o((o)(o)))
                        (o(o)((o)))
                        ((((o)(o))))
                        (((o)((o))))
                        ((o)(((o))))

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

For leaves instead of height we have A185650 aerated, ranked by A358578.
These trees are ranked by A358576.
The ordered version is A358588.
Square trees are counted by A358589, ranked by A358577, ordered A358590.
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.
Cf. A000891, A065097, A342507, A358552, A358581-A358584, 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}]==Depth[#]-1&]],{n,1,10}]

\\ Needs R(n,f) defined in A358589.
seq(n) = {Vec(R(n, (h,p)->polcoef(subst(p, x, x/y), -h, y)), -n)} \\ Andrew Howroyd, Jan 01 2023

Conjectures from Chai Wah Wu, Apr 15 2024: (Start)
a(n) = 5*a(n-1) - 7*a(n-2) - a(n-3) + 8*a(n-4) - 4*a(n-5) for n > 7.
G.f.: x^5*(x^2 - x + 1)/((x - 1)^2*(x + 1)*(2*x - 1)^2). (End)

Terms a(19) and beyond from Andrew Howroyd, Jan 01 2023

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)).

cp's OEIS Frontend

A109082 Depth of rooted tree having Matula-Goebel number n.

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A342507 Number of internal nodes in rooted tree with Matula-Goebel number n.

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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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A358552 Node-height of the rooted tree with Matula-Goebel number n. Number of nodes in the longest path from root to leaf.

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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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A358589 Number of square rooted trees with n nodes.

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A358580 Difference between the number of leaves and the number of internal (non-leaf) nodes in the rooted tree with Matula-Goebel number n.

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