A331872 Number of semi-lone-child-avoiding locally disjoint rooted trees with n vertices.

1, 1, 1, 2, 4, 6, 12, 19, 35, 59, 104, 179, 318, 556, 993, 1772, 3202, 5807, 10643, 19594, 36380, 67915

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

Table of n, a(n) for n=1..22.

David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).

Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

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.

Cf. A000081, A050381, A316696, A316697, A331678, A331679, A331681, A331686, A331687, A331871, A331935.

Sequence in context: A294918 A347474 A370638 * A307067 A358101 A358100

Adjacent sequences: A331869 A331870 A331871 * A331873 A331874 A331875

Keyword

nonn,more

Author

Gus Wiseman, Feb 02 2020

Status

approved