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A358553
Number of internal (non-leaf) nodes in the n-th standard ordered rooted tree.
3
0, 1, 2, 1, 3, 2, 2, 1, 2, 3, 3, 2, 3, 2, 2, 1, 4, 2, 4, 3, 4, 3, 3, 2, 2, 3, 3, 2, 3, 2, 2, 1, 3, 4, 3, 2, 5, 4, 4, 3, 3, 4, 4, 3, 4, 3, 3, 2, 4, 2, 4, 3, 4, 3, 3, 2, 2, 3, 3, 2, 3, 2, 2, 1, 3, 3, 5, 4, 4, 3, 3, 2, 4, 5, 5, 4, 5, 4, 4, 3, 5, 3, 5, 4, 5, 4, 4
OFFSET
1,3
COMMENTS
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.
EXAMPLE
The 89-th standard rooted tree is ((o)o(oo)), and it has 3 internal nodes, so a(89) = 3.
MATHEMATICA
stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]];
srt[n_]:=If[n==1, {}, srt/@stc[n-1]];
Table[Count[srt[n], _[__], {0, Infinity}], {n, 100}]
CROSSREFS
This statistic is counted by A001263, unordered A358575 (reverse A055277).
The unordered version is A342507, firsts A358554.
Other statistics: A358371 (leaves), A358372 (nodes), A358379 (edge-height).
A000081 counts rooted trees, ordered A000108.
Sequence in context: A096852 A096857 A358379 * A303639 A090000 A109082
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 26 2022
STATUS
approved