OFFSET
1,3
COMMENTS
An unlabeled rooted tree is semi-binary if all out-degrees are <= 2, and balanced if all leaves are the same distance from the root. The number of semi-binary trees with n nodes is equal to the number of binary trees with n+1 leaves; see A001190.
LINKS
EXAMPLE
The a(1) = 1 through a(7) = 6 balanced semi-binary rooted trees:
o (o) (oo) ((oo)) (((oo))) ((o)(oo)) ((oo)(oo))
((o)) (((o))) ((o)(o)) ((((oo)))) (((o)(oo)))
((((o)))) (((o)(o))) (((((oo)))))
(((((o))))) ((((o)(o))))
(((o))((o)))
((((((o))))))
MATHEMATICA
saur[n_]:=If[n==1, {{}}, Join@@Table[Select[Union[Sort/@Tuples[saur/@ptn]], SameQ@@Length/@Position[#, {}]&], {ptn, Select[IntegerPartitions[n-1], Length[#]<=2&]}]];
Table[Length[saur[n]], {n, 20}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 08 2018
STATUS
approved