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A358579 Numbers k such that the k-th standard ordered rooted tree has the same number of leaves as internal (non-leaf) nodes.

%I A358579 #6 Nov 26 2022 12:34:55
%S A358579 2,6,7,9,20,22,23,26,27,29,35,41,66,76,78,79,84,86,87,90,91,93,97,102,
%T A358579 103,106,107,109,115,117,130,136,138,139,141,146,153,163,169,193,196,
%U A358579 197,201,226,241,262,263,296,300,302,303,308,310,311,314,315,317
%N A358579 Numbers k such that the k-th standard ordered rooted tree has the same number of leaves as internal (non-leaf) nodes.
%C A358579 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.
%F A358579 A358371(a(n)) = A358553(a(n)).
%e A358579 The terms together with their corresponding rooted trees begin:
%e A358579    2: (o)
%e A358579    6: (o(o))
%e A358579    7: ((oo))
%e A358579    9: ((o)(o))
%e A358579   20: (oo((o)))
%e A358579   22: (o(((o))))
%e A358579   23: (((o)(o)))
%e A358579   26: (o(o(o)))
%e A358579   27: ((o)(o)(o))
%e A358579   29: ((o((o))))
%e A358579   35: (((o))(oo))
%e A358579   41: (((o(o))))
%e A358579   66: (o(o)(((o))))
%e A358579   76: (oo(ooo))
%e A358579   78: (o(o)(o(o)))
%e A358579   79: ((o(((o)))))
%e A358579   84: (oo(o)(oo))
%e A358579   86: (o(o(oo)))
%t A358579 stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
%t A358579 srt[n_]:=If[n==1,{},srt/@stc[n-1]];
%t A358579 Select[Range[100],Count[srt[#],{},{0,Infinity}]==Count[srt[#],_[__],{0,Infinity}]&]
%Y A358579 These ordered trees are counted by A000891.
%Y A358579 The unordered version is A358578, counted by A185650.
%Y A358579 Height instead of leaves: counted by A358588, unordered A358576.
%Y A358579 Height instead of internals: counted by A358590, unordered A358577.
%Y A358579 Standard ordered tree number statistics: A358371, A358372, A358379, A358553.
%Y A358579 A000081 counts rooted trees, ordered A000108.
%Y A358579 A055277 counts trees by nodes and leaves, ordered A001263.
%Y A358579 Cf. A014221, A206487, A358373, A358580, A358587, A358589.
%K A358579 nonn
%O A358579 1,1
%A A358579 _Gus Wiseman_, Nov 25 2022