A358550 Depth of the ordered rooted tree with binary encoding A014486(n).

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

Offset: 1

Author

Gus Wiseman, Nov 22 2022

Keywords

• nonn

Comments

The binary encoding of an ordered tree (A014486) is obtained by replacing the internal left and right brackets with 0's and 1's, thus forming a binary number.

Examples

			The first few rooted trees in binary encoding are:
    0: o
    2: (o)
   10: (oo)
   12: ((o))
   42: (ooo)
   44: (o(o))
   50: ((o)o)
   52: ((oo))
   56: (((o)))
  170: (oooo)
  172: (oo(o))
  178: (o(o)o)
  180: (o(oo))
  184: (o((o)))
		

Crossrefs

Positions of first appearances are A014137.

Leaves of the ordered tree are counted by A057514, standard A358371.

Branches of the ordered tree are counted by A057515.

Edges of the ordered tree are counted by A072643.

The Matula-Goebel number of the ordered tree is A127301.

Positions of 2's are A155587, indices of A020988.

The standard ranking of the ordered tree is A358523.

Nodes of the ordered tree are counted by A358551, standard A358372.

For standard instead of binary encoding we have A358379.

A000108 counts ordered rooted trees, unordered A000081.

A014486 lists all binary encodings.

Cf. A001263, A057122, A358373, A358505, A358524.

Programs

Mathematica

binbalQ[n_]:=n==0||Count[IntegerDigits[n,2],0]==Count[IntegerDigits[n,2],1]&&And@@Table[Count[Take[IntegerDigits[n,2],k],0]<=Count[Take[IntegerDigits[n,2],k],1],{k,IntegerLength[n,2]}];
bint[n_]:=If[n==0,{},ToExpression[StringReplace[StringReplace[ToString[IntegerDigits[n,2]/.{1->"{",0->"}"}],","->""],"} {"->"},{"]]];
Table[Depth[bint[k]]-1,{k,Select[Range[0,1000],binbalQ]}]