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%I #7 Nov 22 2022 11:57:59
%S 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,
%T 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,
%U 3,4,3,3,3,3,4,4,4,4,5,3,3,3,3,4,3,3,3
%N Depth of the ordered rooted tree with binary encoding A014486(n).
%C 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.
%e The first few rooted trees in binary encoding are:
%e 0: o
%e 2: (o)
%e 10: (oo)
%e 12: ((o))
%e 42: (ooo)
%e 44: (o(o))
%e 50: ((o)o)
%e 52: ((oo))
%e 56: (((o)))
%e 170: (oooo)
%e 172: (oo(o))
%e 178: (o(o)o)
%e 180: (o(oo))
%e 184: (o((o)))
%t 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]}];
%t bint[n_]:=If[n==0,{},ToExpression[StringReplace[StringReplace[ToString[IntegerDigits[n,2]/.{1->"{",0->"}"}],","->""],"} {"->"},{"]]];
%t Table[Depth[bint[k]]-1,{k,Select[Range[0,1000],binbalQ]}]
%Y Positions of first appearances are A014137.
%Y Leaves of the ordered tree are counted by A057514, standard A358371.
%Y Branches of the ordered tree are counted by A057515.
%Y Edges of the ordered tree are counted by A072643.
%Y The Matula-Goebel number of the ordered tree is A127301.
%Y Positions of 2's are A155587, indices of A020988.
%Y The standard ranking of the ordered tree is A358523.
%Y Nodes of the ordered tree are counted by A358551, standard A358372.
%Y For standard instead of binary encoding we have A358379.
%Y A000108 counts ordered rooted trees, unordered A000081.
%Y A014486 lists all binary encodings.
%Y Cf. A001263, A057122, A358373, A358505, A358524.
%K nonn
%O 1,2
%A _Gus Wiseman_, Nov 22 2022
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