I discovered an interesting relationship a few days ago

(and I got a chance to be my own collaborator).

I was generating rooted trees of nonempty sets with n points
(i.e. rooted trees whose nodes are sets of 1 or more points.)

So I worked out the first few terms

1,2,5,13,37,108,332

I deleted the initial "1" and sent it to the server and got:

%I A029856
%S A029856 2,2,5,13,37,108,332,1042,3360,11019,36722,123875,422449,1453553,
%T A029856 5040816,17599468,61814275,218252584,774226549,2758043727,
%U A029856 9862357697,35387662266,127374191687,459783039109,1664042970924
%N A029856 Rooted trees with 2 types of leaves.
%O A029856 1,1
%K A029856 nonn,easy,eigen
%F A029856 Sequence is its own Euler transform.
%H A029856 <a href="http://www.research.att.com/~njas/sequences/transforms.html">Transforms</a>
%A A029856 Christian Bower (bowerc@usa.net)
 
I generated several more terms and saw that the 2 sequences were very likely
the same (except for the first term).

So I set out to find a correspondence between the 2 sequences.

It turns out to be rather simple:

To turn a rooted tree with 2 colored leaves into a rooted tree of
nonempty sets, take the white leaves and join them into the parent
node, so that a node with n white leaves as children becomes a
set node with n+1 points.

Similarly a node with is a set of at least 2 points can be turned
into a node with n-1 white leaves as children. The leaves which are
sets of 1 point become the black nodes.

The bijection does not work for a one node rooted tree because
it has a leaf which has no parent.

This method also relates (non-rooted) trees with 2-colored nodes to
trees of nonempty sets as long as they have at least 3 nodes (or
points.)

The tree sequences begin
1,2,3,7,14,35,85,231,... for trees of nonempty sets
and
2,3,3,7,14,35,85,231,... for trees with 2-colored nodes.

                                   ___
                                  | . |
                                  |___|
                              __   _|_
                             | .| | ..|
         B W                 |__| |___|
         |/                   _|_  _|_
 B W W W .                   |...|| ..|
  \|/  |/         <--->      |___||___|
   .   .                        _\/_
    \ /                         | .|
     R                          |__|

Christian