OFFSET
1,2
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.
EXAMPLE
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)))
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]}]
CROSSREFS
Positions of first appearances are A014137.
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.
The standard ranking of the ordered tree is A358523.
For standard instead of binary encoding we have A358379.
A014486 lists all binary encodings.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 22 2022
STATUS
approved