Monotonic trees

[franktaw at netscape.net]

Define a rooted tree is monotonic if the out-degree (number of 
children)
of every node is greater than or equal to the out-degree of any of its
children.  How many monotonic trees are there with n nodes?

Assuming unordered trees, I believe the sequence starts:

1,1,2,3,6,10,21;

and for ordered trees:

1,1,2,4,10,25,68.

I don't believe that either of these is in the OEIS, although this 
isn't
enough terms to tell for certain.

Can somebody extend either or both of these?

(One could make similar definition with out-degree increasing from the
root, but you have to exclude leaf nodes from consideration.  This is
much cleaner.)

Franklin T. Adams-Watters

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