  Let $f:\mathbb{C}^2 \to \mathbb{C}$ be a polynomial map. Let $\mathbb{C}^2
\subset X$ be a compactification of $\mathbb{C}^2$ where $X$ is a smooth
rational compact surface and such that there exists a morphism of varieties
$\Phi :X\to \mathbb{P}^1$ which extends $f$. Put $\mathcal{D}=X\setminus
\mathbb{C}^2$; $\mathcal{D}$ is a curve whose irreducible components are smooth
rational compact curves and all its singularities are ordinary double points.
The dual graph of $\mathcal{D}$ is a tree. We are interested in this tree, and
we analyse its complexity in terms of the genus of the generic fiber of $f$.
