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

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

1,1

Comments

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.

Links

Table of n, a(n) for n=1..57.

Formula

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

Example

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)))

Mathematica

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}]&]

Crossrefs

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.

Sequence in context: A200926 A047277 A308198 * A341437 A327985 A189465

Adjacent sequences: A358576 A358577 A358578 * A358580 A358581 A358582

Keyword

nonn

Author

Gus Wiseman, Nov 25 2022

Status

approved