A209638 -id:A209638 - cp's OEIS frontend

A243497 A243496 sorted into ascending order, with duplicates removed.

1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 17, 19, 23, 25, 27, 31, 32, 49, 53, 59, 64, 67, 81, 83, 97, 103, 121, 125, 127, 128, 131, 169, 227, 241, 243, 256, 277, 289, 311, 331, 343, 361, 419, 431, 509, 512, 529, 563, 661, 691, 709, 719, 739, 841, 961, 1009, 1024, 1433, 1523, 1619, 1681, 1787, 1849, 1879, 2063

Offset: 0

Antti Karttunen, Jun 07 2014

nonn

Matula-codes for trees which are almost "uniform", but which allow cases like 169, 841, 1009, 1681, 1849, ... where there is a special relation between prime index and the exponent. (Cf. the comments at A243496).
Differs from A214577 for the first time at n=31, where A214577(31)=227, while here we have 169 at that position, because it corresponds exactly to that "dual" case mentioned in A057546, in excess to those mentioned in A003238. Note that 169 = 13*13 = p_{2*3}^2.
a(0) = 1 stands for the empty tree.

Cf. A214577 (a subsequence), A209638, A243494.

A358731 Matula-Goebel numbers of rooted trees whose number of nodes is one more than their node-height.

Original entry on oeis.org

4, 6, 7, 10, 13, 17, 22, 29, 41, 59, 62, 79, 109, 179, 254, 277, 293, 401, 599, 1063, 1418, 1609, 1787, 1913, 2749, 4397, 8527, 10762, 11827, 13613, 15299, 16519, 24859, 42043, 87803

Offset: 1

Gus Wiseman, Dec 01 2022

These are paths with a single extra leaf growing from them.
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:
    4: (oo)
    6: (o(o))
    7: ((oo))
   10: (o((o)))
   13: ((o(o)))
   17: (((oo)))
   22: (o(((o))))
   29: ((o((o))))
   41: (((o(o))))
   59: ((((oo))))
   62: (o((((o)))))
   79: ((o(((o)))))
  109: (((o((o)))))
  179: ((((o(o)))))
  254: (o(((((o))))))
  277: (((((oo)))))
  293: ((o((((o))))))
  401: (((o(((o))))))
  599: ((((o((o))))))

These trees are counted by A289207.
Positions of 1's in A358729.
A000081 counts rooted trees, ordered A000108.
A034781 counts rooted trees by nodes and height.
A055277 counts rooted trees by nodes and leaves.
MG differences: A358580, A358724, A358726, A358729.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
MG core: A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A206487, A209638, A316321, A358576, A358577, A358592, A358725.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Count[MGTree[#],_,{0,Infinity}]==Depth[MGTree[#]]&]

A358727 Matula-Goebel numbers of rooted trees with greater number of leaves (width) than node-height.

Original entry on oeis.org

8, 16, 24, 28, 32, 36, 38, 42, 48, 49, 53, 54, 56, 57, 63, 64, 72, 76, 80, 81, 84, 96, 98, 104, 106, 108, 112, 114, 120, 126, 128, 131, 133, 136, 140, 144, 147, 148, 152, 156, 159, 160, 162, 168, 171, 172, 178, 180, 182, 184, 189, 190, 192, 196, 200, 204, 208

Offset: 1

Gus Wiseman, Dec 01 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:
   8: (ooo)
  16: (oooo)
  24: (ooo(o))
  28: (oo(oo))
  32: (ooooo)
  36: (oo(o)(o))
  38: (o(ooo))
  42: (o(o)(oo))
  48: (oooo(o))
  49: ((oo)(oo))
  53: ((oooo))
  54: (o(o)(o)(o))
  56: (ooo(oo))
  57: ((o)(ooo))
  63: ((o)(o)(oo))
  64: (oooooo)
  72: (ooo(o)(o))
  76: (oo(ooo))

Gus Wiseman, The first 64 rooted trees with greater width than height.

Positions of negative terms in A358726.
These trees are counted by A358728.
Differences: A358580, A358724, 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, A209638, A358576, A358577, A358578, A358587, A358730.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Depth[MGTree[#]]-1

A358723 Number of n-node rooted trees of edge-height equal to their number of leaves.

Original entry on oeis.org

0, 1, 0, 2, 1, 6, 7, 26, 43, 135, 276, 755, 1769, 4648, 11406, 29762, 75284, 195566, 503165, 1310705, 3402317, 8892807, 23231037, 60906456, 159786040, 420144405, 1105673058, 2914252306, 7688019511, 20304253421, 53667498236, 141976081288, 375858854594, 995728192169

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 a(1) = 0 through a(7) = 7 trees:
  .  (o)  .  ((oo))  ((o)(o))  (((ooo)))  (((o))(oo))
             (o(o))            ((o(oo)))  (((o)(oo)))
                               ((oo(o)))  ((o)((oo)))
                               (o((oo)))  ((o)(o(o)))
                               (o(o(o)))  ((o(o)(o)))
                               (oo((o)))  (o((o)(o)))
                                          (o(o)((o)))

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

For internals instead of leaves: A011782, ranked by A209638.
For internals instead of edge-height: A185650 aerated, ranked by A358578.
For node-height: A358589 (square trees), 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 internals, ordered A090181.
Cf. A065097, A109082, A109129, A342507, A358552, A358587, A358591, A358728.

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}]==Depth[#]-2&]],{n,1,10}]

\\ Needs R(n,f) defined in A358589.
seq(n) = {Vec(R(n, (h,p)->polcoef(p,h-1,y)), -n)} \\ Andrew Howroyd, Jan 01 2023

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

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A243497 A243496 sorted into ascending order, with duplicates removed.

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A358731 Matula-Goebel numbers of rooted trees whose number of nodes is one more than their node-height.

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A358727 Matula-Goebel numbers of rooted trees with greater number of leaves (width) than node-height.

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A358723 Number of n-node rooted trees of edge-height equal to their number of leaves.

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