A358375 Numbers k such that the k-th standard ordered rooted tree is binary.

1, 4, 18, 25, 137, 262146, 393217, 2097161, 2228225

Offset: 1

Author

Gus Wiseman, Nov 14 2022

Keywords

• nonn
• more

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.

Examples

			The initial terms and their corresponding trees:
       1: o
       4: (oo)
      18: ((oo)o)
      25: (o(oo))
     137: ((oo)(oo))
  262146: (((oo)o)o)
  393217: (o((oo)o))
		

Links

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

Crossrefs

The unordered version is A111299, counted by A001190

These trees are counted by A126120.

A000081 counts unlabeled rooted trees, ranked by A358378.

A358371 and A358372 count leaves and nodes in standard ordered rooted trees.

Cf. A001263, A004249, A005043, A063895, A245824, A284778, A358373, A358374, A358376, A358377.

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
Select[Range[1000],FreeQ[srt[#],[_]?(Length[#]!=2&)]&]