cp's OEIS Frontend

A358371 Number of leaves in the n-th standard ordered rooted tree.

%I A358371 #10 Nov 14 2022 09:57:35
%S A358371 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 A358371 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 A358371 3,4,4,5,3,3,3,4,3,4,4,5,2,4,3,4,3,4
%N A358371 Number of leaves in the n-th standard ordered rooted tree.
%C A358371 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 A358371 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 A358371 The standard ordered rooted tree ranking begins:
%e A358371   1: o        10: (((o))o)   19: (((o))(o))
%e A358371   2: (o)      11: ((o)(o))   20: (((o))oo)
%e A358371   3: ((o))    12: ((o)oo)    21: ((o)((o)))
%e A358371   4: (oo)     13: (o((o)))   22: ((o)(o)o)
%e A358371   5: (((o)))  14: (o(o)o)    23: ((o)o(o))
%e A358371   6: ((o)o)   15: (oo(o))    24: ((o)ooo)
%e A358371   7: (o(o))   16: (oooo)     25: (o(oo))
%e A358371   8: (ooo)    17: ((((o))))  26: (o((o))o)
%e A358371   9: ((oo))   18: ((oo)o)    27: (o(o)(o))
%e A358371 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 A358371 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A358371 srt[n_]:=If[n==1,{},srt/@stc[n-1]];
%t A358371 Table[Count[srt[n],{},{0,Infinity}],{n,100}]
%Y A358371 The triangle counting trees by this statistic is A001263, unordered A055277.
%Y A358371 The version for unordered trees is A109129, nodes A061775, edges A196050.
%Y A358371 The nodes are counted by A358372.
%Y A358371 A000081 counts unordered rooted trees, ranked by A358378.
%Y A358371 A000108 counts ordered rooted trees.
%Y A358371 A358374 ranks ordered identity trees, counted by A032027.
%Y A358371 A358375 ranks ordered binary trees, counted by A126120
%Y A358371 Cf. A004249, A005043, A063895, A187306, A284778, A358373, A358376, A358377.
%K A358371 nonn
%O A358371 1,4
%A A358371 _Gus Wiseman_, Nov 13 2022