if $i=2$. $\nabla_{p^i}^{(k)}S$ stands for the $k^{th}$ derivative of $S$ w.r.t. $p^i$ evaluated at $(p_1+p_2,p_3,x)$ if $i=1$ and at $(p_1,p_2+p_3)$ if $i=2$.

Let $i=1,2$, $t\in[RT]$. The elementary generating function $S^i(t)$ of $S\in C^\infty(B_2)[[h]]$ is defined recursively as follows,

1.  $S^1(\circ) =  S(p_1,p_2,x)$ ,$S^1(\bullet) =  S(p_1+p_2,p_3,x)$

2.  $S^2(\circ) =  S(p_2,p_3,x)$ ,$S^2(\bullet) =  S(p_1,p_2+p_3,x)$

3.  $S^i(t) = \nabla_{p^i}^{(m+1)}S(D^iS(t_1),\dots,D^iS(t_m))$ if $t= [t_1,\dots,t_m]_\bullet$

4.  $S^i(t) = \nabla_x^{(m+1)}S(D^iS(t_1),\dots,D^iS(t_m))$ if $t= [t_1,\dots,t_m]_\circ$

with the same notation as above.

Some examples are given in the following table:

  --------- ----------------------------------- ------------------------------------------------------------- -------------------------------------------------------------
   Diagram               Notation                                            ED                                                            EGF
                                                                                                              
                    $[\bullet]_\circ $                          $\nabla^{(2)}_x S\nabla_p S $                                    $\nabla_x S\nabla_p S$
                                                                                                              
                  $[\circ,\circ]_\bullet$                $ \nabla^{(3)}_pS(\nabla_x S, \nabla_x S) $                    $\nabla^{(2)}_pS(\nabla_x S, \nabla_x S)$
                                                                                                              
                                                                                                              
             $[\bullet,[\circ]_\bullet]_\circ$   $ \nabla^{(3)}_xS(\nabla_p S,\nabla^{(2)}_p S\nabla_x S ) $   $ \nabla^{(2)}_xS(\nabla_p S,\nabla^{(2)}_p S\nabla_x S ) $
                                                                                                              
  --------- ----------------------------------- ------------------------------------------------------------- -------------------------------------------------------------

Remark that for EGF it is not important which vertex is the root. This is not the case for ED. Let us be more precise.

Let $u = [u_1,\dots,u_k], v = [v_1,\dots,v_l]\in [RT]$. We denote by $$\begin{aligned}
u\circ v &=& [u_1,\dots,u_k,v]\\
v\circ u &=& [v_1,\dots,v_l,u]\\\end{aligned}$$ the Butcher product. We have not written the obvious conditions on the $u_i$ and $v_i$ so that the product remains bipartite.

We consider the minimal equivalence relation on 