OFFSET
1,1
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) = 2 through a(4) = 24 trees:
o (oo) (ooo) (oooo)
(o) (o(o)) (o(oo)) (o(ooo))
((o)(o)) (oo(o)) (oo(oo))
(o(o)(o)) (ooo(o))
(o(o(o))) ((oo)(oo))
((o)(o)(o)) (o(o(oo)))
(o((o)(o))) (o(oo(o)))
((o)((o)(o))) (oo(o)(o))
(oo(o(o)))
(o(o)(o)(o))
(o(o(o)(o)))
(o(o(o(o))))
(oo((o)(o)))
((o)(o)(o)(o))
((o(o))(o(o)))
((oo)((o)(o)))
(o((o)(o)(o)))
(o(o)((o)(o)))
(o(o((o)(o))))
((o)((o)(o)(o)))
((o)(o)((o)(o)))
(o((o)((o)(o))))
(((o)(o))((o)(o)))
((o)((o)((o)(o))))
MATHEMATICA
disjointQ[u_]:=Apply[And, Outer[#1==#2||Intersection[#1, #2]=={}&, u, u, 1], {0, 1}];
slaurt[n_]:=If[n==1, {o, {o}}, Join@@Table[Select[Union[Sort/@Tuples[slaurt/@ptn]], disjointQ[Select[#, !AtomQ[#]&]]&], {ptn, Rest[IntegerPartitions[n]]}]];
Table[Length[slaurt[n]], {n, 8}]
CROSSREFS
Not requiring local disjointness gives A050381.
The non-semi version is A316697.
The same trees counted by number of vertices are A331872.
The Matula-Goebel numbers of these trees are A331873.
Lone-child-avoiding rooted trees counted by leaves are A000669.
Semi-lone-child-avoiding rooted trees counted by vertices are A331934.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Feb 02 2020
STATUS
approved