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%I #15 Feb 01 2020 07:08:24
%S 1,0,1,1,2,3,6,9,16,26,45,72,124,201,341,561,947,1571,2651,4434,7496,
%T 12631,21423,36332,61910,105641,180924,310548,534713,923047
%N Number of lone-child-avoiding locally disjoint unlabeled rooted trees with n vertices.
%C First differs from A320268 at a(11) = 45, A320268(11) = 44.
%C A rooted tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other unequal child of the same vertex. Lone-child-avoiding means there are no unary branchings.
%H David Callan, <a href="http://arxiv.org/abs/1406.7784">A sign-reversing involution to count labeled lone-child-avoiding trees</a>, arXiv:1406.7784 [math.CO], (30-June-2014).
%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vS1zCO9fgAIe5rGiAhTtlrOTuqsmuPos2zkeFPYB80gNzLb44ufqIqksTB4uM9SIpwlvo-oOHhepywy/pub">Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.</a>
%e The a(1) = 1 through a(9) = 16 trees (empty column indicated by dot):
%e o . (oo) (ooo) (oooo) (ooooo) (oooooo) (ooooooo) (oooooooo)
%e (o(oo)) (o(ooo)) (o(oooo)) (o(ooooo)) (o(oooooo))
%e (oo(oo)) (oo(ooo)) (oo(oooo)) (oo(ooooo))
%e (ooo(oo)) (ooo(ooo)) (ooo(oooo))
%e ((oo)(oo)) (oooo(oo)) (oooo(ooo))
%e (o(o(oo))) (o(o(ooo))) (ooooo(oo))
%e (o(oo)(oo)) ((ooo)(ooo))
%e (o(oo(oo))) (o(o(oooo)))
%e (oo(o(oo))) (o(oo(ooo)))
%e (o(ooo(oo)))
%e (oo(o(ooo)))
%e (oo(oo)(oo))
%e (oo(oo(oo)))
%e (ooo(o(oo)))
%e (o((oo)(oo)))
%e (o(o(o(oo))))
%t disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
%t strut[n_]:=If[n==1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[strut/@c]]]/@Rest[IntegerPartitions[n-1]],disjointQ]];
%t Table[Length[strut[n]],{n,10}]
%Y The enriched version is A316696.
%Y The Matula-Goebel numbers of these trees are A331871.
%Y The non-locally disjoint version is A001678.
%Y These trees counted by number of leaves are A316697.
%Y The semi-lone-child-avoiding version is A331872.
%Y Cf. A000081, A000669, A005804, A060356, A141268, A300660, A316471, A316473, A316694, A316495, A319312, A330465, A331679, A331681, A331683.
%K nonn,more
%O 1,5
%A _Gus Wiseman_, Jan 25 2020
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