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A358728
Number of n-node rooted trees whose node-height is less than their number of leaves.
3
0, 0, 0, 1, 1, 5, 10, 30, 76, 219, 582, 1662, 4614, 13080, 36903, 105098, 298689, 852734, 2434660, 6964349, 19931147, 57100177, 163647811, 469290004, 1346225668, 3863239150, 11089085961, 31838349956, 91430943515, 262615909503, 754439588007, 2167711283560
OFFSET
1,6
COMMENTS
Node-height is the number of nodes in the longest path from root to leaf.
LINKS
EXAMPLE
The a(1) = 0 through a(7) = 10 trees:
. . . (ooo) (oooo) (ooooo) (oooooo)
((oooo)) ((ooooo))
(o(ooo)) (o(oooo))
(oo(oo)) (oo(ooo))
(ooo(o)) (ooo(oo))
(oooo(o))
((o)(ooo))
((oo)(oo))
(o(o)(oo))
(oo(o)(o))
MATHEMATICA
art[n_]:=If[n==1, {{}}, Join@@Table[Select[Tuples[art/@c], OrderedQ], {c, Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[art[n], Depth[#]-1<Count[#, {}, {-2}]&]], {n, 1, 10}]
PROG
(PARI) \\ Needs R(n, f) defined in A358589.
seq(n) = {Vec(R(n, (h, p)->sum(j=h+1, n-1, polcoef(p, j, y))), -n)} \\ Andrew Howroyd, Jan 01 2023
CROSSREFS
These trees are ranked by A358727.
For internals instead of node-height we have A358581, ordered A358585.
The case of equality is A358589 (square trees), ranked by A358577.
A000081 counts rooted trees, ordered A000108.
A034781 counts rooted trees by nodes and height, ordered A080936.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
Sequence in context: A048010 A002571 A077916 * A373568 A189315 A056422
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 29 2022
EXTENSIONS
Terms a(19) and beyond from Andrew Howroyd, Jan 01 2023
STATUS
approved