[seqfan] Sequence of 2-rooted trees

[Richard J. Mathar]

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.