%I #10 Nov 14 2022 09:57:35
%S 1,1,1,2,1,2,2,3,2,2,2,3,2,3,3,4,1,3,2,3,2,3,3,4,3,3,3,4,3,4,4,5,2,2,
%T 3,4,2,3,3,4,3,3,3,4,3,4,4,5,2,4,3,4,3,4,4,5,4,4,4,5,4,5,5,6,2,3,2,3,
%U 3,4,4,5,3,3,3,4,3,4,4,5,2,4,3,4,3,4
%N Number of leaves in the n-th standard ordered rooted tree.
%C 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.
%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>
%e The standard ordered rooted tree ranking begins:
%e 1: o 10: (((o))o) 19: (((o))(o))
%e 2: (o) 11: ((o)(o)) 20: (((o))oo)
%e 3: ((o)) 12: ((o)oo) 21: ((o)((o)))
%e 4: (oo) 13: (o((o))) 22: ((o)(o)o)
%e 5: (((o))) 14: (o(o)o) 23: ((o)o(o))
%e 6: ((o)o) 15: (oo(o)) 24: ((o)ooo)
%e 7: (o(o)) 16: (oooo) 25: (o(oo))
%e 8: (ooo) 17: ((((o)))) 26: (o((o))o)
%e 9: ((oo)) 18: ((oo)o) 27: (o(o)(o))
%e For example, the 25th ordered tree is (o,(o,o)) because the 24th composition is (1,4) and the 3rd composition is (1,1). Hence a(25) = 3.
%t stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t srt[n_]:=If[n==1,{},srt/@stc[n-1]];
%t Table[Count[srt[n],{},{0,Infinity}],{n,100}]
%Y The triangle counting trees by this statistic is A001263, unordered A055277.
%Y The version for unordered trees is A109129, nodes A061775, edges A196050.
%Y The nodes are counted by A358372.
%Y A000081 counts unordered rooted trees, ranked by A358378.
%Y A000108 counts ordered rooted trees.
%Y A358374 ranks ordered identity trees, counted by A032027.
%Y A358375 ranks ordered binary trees, counted by A126120
%Y Cf. A004249, A005043, A063895, A187306, A284778, A358373, A358376, A358377.
%K nonn
%O 1,4
%A _Gus Wiseman_, Nov 13 2022