[seqfan] A248869 trees count

[Kevin Ryde]

>From trying some code, it looks like A248869 counts how many rooted
unlabelled trees have isomorphic sibling subtrees in sets of at most 2.
So a vertex can have 1 sibling of the same (isomorphic) subtree, but no
more than that.  (Reckoning root at the top, and "subtree" as the vertex
and everything below.)

Is there a name for trees with limited symmetry like this?  Is there a
name for their automorphism groups?

Tree automorphism groups as usual are symmetric group S(n) in direct
products, and wreath products TreeGroup wr S(n) when nesting.  With n<=2
for A248869, they are some of the 2-groups (order 2^k).  Or n<=4 should
be some of the solvable groups.  If I have my jargon right ...