[seqfan] Re: Sequence of 2-rooted trees

[Neil Sloane]

Richard,  That's a surprise, a new "trees" sequence.  It does seem to be
new - please submit it right away!

Cayley studied centered trees, and bi-centered trees, but as far as I know
he never looked at bi-rooted trees.

I think "Bi-rooted trees on n nodes" would be a good name.

You were looking at the case when the nodes are unlabeled, but there is
also the labeled case, which maybe begins 0,1,9,96,... and also seems to be
new

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com


On Mon, Apr 30, 2018 at 9:41 AM, Richard J. Mathar <mathar at mpia-hd.mpg.de>
wrote:

> Is the number of 2-rooted trees (unlabeled, undirected) somewhere
> mentioned in the OEIS? My guess is that this sequence
> starts 0, 1, 2, 6, 15, 43, 116, 329, 918, 2609, with offset 1.
> The concept means to mark two nodes of a tree with some (the same)
> mark, generalizing the rooted trees  (A000081) which have only one node
> marked.
>
> The 2-rooted tree has a unique path/bridge between the two nodes, which may
> divert into branches; the other vertices are a rooted tree of one root,
> and a rooted tree of the other root.  The middle section (bridge between
> the roots) may be as simple as a single edge, but is in general also a
> (2-rooted) tree.
>
> The forests with 2 rooted trees, column 2 in A033185, are a special
> (limited) case, because if we connect the roots of two rooted trees by
> a single edge, we have constructed a 2-rooted tree.
>
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