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A358371
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Number of leaves in the n-th standard ordered rooted tree.
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26
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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, 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, 3, 4, 4, 5, 3, 3, 3, 4, 3, 4, 4, 5, 2, 4, 3, 4, 3, 4
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OFFSET
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1,4
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COMMENTS
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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.
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LINKS
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Table of n, a(n) for n=1..86.
Gus Wiseman, Statistics, classes, and transformations of standard compositions
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EXAMPLE
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The standard ordered rooted tree ranking begins:
1: o 10: (((o))o) 19: (((o))(o))
2: (o) 11: ((o)(o)) 20: (((o))oo)
3: ((o)) 12: ((o)oo) 21: ((o)((o)))
4: (oo) 13: (o((o))) 22: ((o)(o)o)
5: (((o))) 14: (o(o)o) 23: ((o)o(o))
6: ((o)o) 15: (oo(o)) 24: ((o)ooo)
7: (o(o)) 16: (oooo) 25: (o(oo))
8: (ooo) 17: ((((o)))) 26: (o((o))o)
9: ((oo)) 18: ((oo)o) 27: (o(o)(o))
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.
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MATHEMATICA
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stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
srt[n_]:=If[n==1, {}, srt/@stc[n-1]];
Table[Count[srt[n], {}, {0, Infinity}], {n, 100}]
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CROSSREFS
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The triangle counting trees by this statistic is A001263, unordered A055277.
The version for unordered trees is A109129, nodes A061775, edges A196050.
The nodes are counted by A358372.
A000081 counts unordered rooted trees, ranked by A358378.
A000108 counts ordered rooted trees.
A358374 ranks ordered identity trees, counted by A032027.
A358375 ranks ordered binary trees, counted by A126120
Cf. A004249, A005043, A063895, A187306, A284778, A358373, A358376, A358377.
Sequence in context: A064122 A323424 A334098 * A263922 A057526 A033265
Adjacent sequences: A358368 A358369 A358370 * A358372 A358373 A358374
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KEYWORD
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nonn
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AUTHOR
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Gus Wiseman, Nov 13 2022
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STATUS
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approved
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