Let be B(z) the exponential generating function for the number $b_n$ of different rooted unordered binary trees with exactly n leaves labeled only at their leaves (so the internal nodes are unlabeled). Then it's a well know result that $b_n = 1\cdot3\cdot\ldots\cdot(2n-3)=(2n-3)!!$ (see eg. Schröder "Vier combinatorische Probleme"). The EGF satisfies the recursion
$B(z) = z + \frac{1}{2}B(z)^2$.
There is one tree with one leave and every tree is built up from two (smaller) trees where the order doesn't matter (because the trees are unplane).
My question:
So $B(z) = z + z^2 + 3z^3 + 15z^4 + 105z^5 + ...$
And therefore $z + \frac{1}{2}B(z)^2 = z + \frac{1}{2}(z^2 + 2z^3 + 7z^4 + 36z^5 + ...) \neq B(z)$
So the recursion is not satisfied? Where is my mistake?
Thanks a lot!