OFFSET
1,4
COMMENTS
A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf.
Locally disjoint means no child of any vertex has branches overlapping the branches of any other (inequivalent) child of the same vertex.
LINKS
David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
EXAMPLE
The a(1) = 1 through a(8) = 19 trees:
o (o) (oo) (ooo) (oooo) (ooooo) (oooooo) (ooooooo)
(o(o)) (o(oo)) (o(ooo)) (o(oooo)) (o(ooooo))
(oo(o)) (oo(oo)) (oo(ooo)) (oo(oooo))
((o)(o)) (ooo(o)) (ooo(oo)) (ooo(ooo))
(o(o)(o)) (oooo(o)) (oooo(oo))
(o(o(o))) ((oo)(oo)) (ooooo(o))
(o(o(oo))) (o(o(ooo)))
(o(oo(o))) (o(oo)(oo))
(oo(o)(o)) (o(oo(oo)))
(oo(o(o))) (o(ooo(o)))
((o)(o)(o)) (oo(o(oo)))
(o((o)(o))) (oo(oo(o)))
(ooo(o)(o))
(ooo(o(o)))
(o(o)(o)(o))
(o(o(o)(o)))
(o(o(o(o))))
(oo((o)(o)))
((o)((o)(o)))
MATHEMATICA
disjointQ[u_]:=Apply[And, Outer[#1==#2||Intersection[#1, #2]=={}&, u, u, 1], {0, 1}];
strutsemi[n_]:=If[n==1, {{}}, If[n==2, {{{}}}, Select[Join@@Function[c, Union[Sort/@Tuples[strutsemi/@c]]]/@Rest[IntegerPartitions[n-1]], disjointQ]]];
Table[Length[strutsemi[n]], {n, 8}]
CROSSREFS
Not requiring lone-child-avoidance gives A316473.
The non-semi version is A331680.
The Matula-Goebel numbers of these trees are A331873.
The same trees counted by number of leaves are A331874.
Not requiring local disjointness gives A331934.
Lone-child-avoiding rooted trees are A001678.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Feb 02 2020
STATUS
approved