$[RT]$ such that $u\circ v \sim  v\circ u$.

**Properties of this relation:**

It is clear that

1.  two topological rooted trees are equivalent if it is possible to pass from one to the other by changing the root. More precisely: $t,t'\in[RT]$, $t\sim t'$ iff there exists a representative $(E,V,r)$ of $t$ and a representative $(E',V',r')$ of $t'$ and a vertex $r''\in V$ such that $(E,V,r'')$ and $(E',V',r')$ are isomorphic rooted trees.

2.  the quotient of $[RT]$ by this equivalence relation is exactly $[T]$.

3.  it follows immediately from the definition $S^i(t) = S^i(t')$ if $t\sim t'$ for $i=1,2$.

Then, it makes sense to define the EGF on bipartite trees.

Let $S\in C^\infty (B_2)[[h]]$ and $t=(V_t,E_t)\in T$. Then

$$S^1(t) := \sum_{\beta:E_t\rightarrow \{ 1,\dots,d\}}\prod_{v\in V_t}\Big[\Big(
\prod_{\substack{e\in E_t\\e=\{*,v\}}}D^{1,\omega(v)}_{\beta(e)} \Big)S \Big]$$ where $$D^{1,\bullet}_{\beta(e_1)}\dots D^{1,\bullet}_{\beta(e_k)}S := \frac{\partial^k S}{\partial p^{1}_{\beta(e_1)}\dots 
\partial p^{1}_{\beta(e_k)}}(p_1+p_2,p_3,x)$$

$$D^{1,\circ}_{\beta(e_1)}\dots D^{1,\circ}_{\beta(e_k)}S := \frac{\partial^k S}{\partial x^{\beta(e_1)}\dots \partial x^{\beta(e_k)}}(p_1,p_2+p_3,x)$$ and correspondingly for $S^2(t)$.

It is clear that this new definition of $S^i(t)$ is equivalent to the previously introduced recursive one. This definition is however better if we want to deal with the fact that $S$ is a formal series. Namely we immediately get the relation

$$\begin{aligned}
h^{|t|}S^i(t) & = & h^{|t|}\sum_{\beta:E_t\rightarrow\{1,\dots,d\}}\prod_{v\in V_t}\Big[\big(          \prod_{\substack{e\in E_t\\e=\{*,v\}}}D^{i,\omega(v)}_{\beta(e)}\big) \big\{ 