(v))=\omega(v)$.]{}

The following table summarizes some notations we will use in the sequel.

  -------------- ---------------------------------------------
       $T$       the set of bipartite trees
       $RT$      the set of rooted bipartite trees
    $RT_\circ$   the set of elements of $RT$ with white root
   $RT_\bullet$  the set of elements of $RT$ with black root
  -------------- ---------------------------------------------

$[A]$ : the set of equivalence classes of graphs in $A$ (ex: $[RT]$). They are called **topological** “$A$" trees.

The elements of $[RT]$ can be described recursively as follows

1.  [$\circ,\bullet\in [RT]$]{}

2.  [if $t_1,\dots,t_m\in [RT_\circ]$, then the tree $[t_1,\dots,t_m]_\bullet\in [RT]$ where $[t_1,\dots,t_m]_\bullet$ is defined by connecting the roots of $t_1,\dots,t_m$ with $\bullet$ and saying that $\bullet$ is the new root. And the same if we interchange $\circ$ and $\bullet$. ]{}

Now with the help of this recursive description of topological rooted trees we define elementary differentials and elementary generating functions.

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

1.  $D^iS(\circ) = \nabla_x S$ , $D^iS(\bullet) = \nabla_{p^i} S$

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

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

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