Denote the e.g.f. for the number of (unordered) rooted labeled trees on $n$ nodes by
$$\Phi(x)=\sum\_{n\geq1}\frac{n^{n-1}}{n!}x^n.$$
And, the related series $\Psi(x)=\sum\_{n\geq1}\frac{n^n}{n!}x^n$. Designate the operator $D:=x\frac{d}{dx}$. Some properties:


(1) $\Psi=D\Phi$;


(2) both $\Phi$ and $\Psi$ have radius of convergence $\frac1e$;


(3) (from Lagrange) $\Phi(x)$ is the inverse of $x(v)=ve^{-v}$;


(4) $(1-\Phi)(1+\Psi)=1$;


(5) $\Phi^k(x)=k\sum\_{n\geq j}\frac{n^{n-k-1}}{(n-k)!}x^n$.



> 
> **Questions.**
> 
> 
> (a) Is there anything interesting that can be said about the algebra $\mathcal{A}=\mathbb{Q}[\Phi,\Psi]$?
> 
> 
> (b) In view of (4) above, what about the quotient of $\mathcal{A}$ by the relation $\Phi\Psi-\Psi+\Phi$?
> 
> 
> 



an algebra generated by some known series
This has been considered by Dimitri Zvonkine, see his article "An algebra of power series arising in the
intersection theory of moduli spaces of curves
and in the enumeration of ramified coverings
of the sphere" (<https://arxiv.org/pdf/math/0403092.pdf>).


