A358459 Numbers k such that the k-th standard ordered rooted tree is balanced (counted by A007059).

1, 2, 3, 4, 5, 8, 9, 11, 16, 17, 32, 35, 37, 41, 43, 64, 128, 129, 137, 139, 163, 169, 171, 256, 257, 293, 512, 515, 529, 547, 553, 555, 641, 649, 651, 675, 681, 683, 1024, 1025, 2048, 2053, 2057, 2059, 2177, 2185, 2187, 2211, 2217, 2219, 2305, 2341, 2563

Offset: 1

Author

Gus Wiseman, Nov 19 2022

Keywords

• nonn

Comments

An ordered tree is balanced if all leaves have the same distance from the root.

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 terms together with their corresponding ordered trees begin:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   8: (ooo)
   9: ((oo))
  11: ((o)(o))
  16: (oooo)
  17: ((((o))))
  32: (ooooo)
  35: ((oo)(o))
  37: (((o))((o)))
  41: ((o)(oo))
  43: ((o)(o)(o))
		

Links

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

Crossrefs

These trees are counted by A007059.

The unordered version is A184155, counted by A048816.

A000108 counts ordered rooted trees, unordered A000081.

A358379 gives depth of standard ordered trees.

Cf. A003238, A004249, A032027, A244925, A290822, A318185, A358373-A358378.

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[100],SameQ@@Length/@Position[srt[#],{}]&]