  William Thurston (1946-2012) gave a combinatorial characterization for
generic branched self-coverings of the two-sphere by associating a planar graph
to them 10.48550/arXiv.1502.04760. By generalizing the notion of local
balancing, the author extends the Thurston result to encompass any branched
covering of the two-sphere. As an application, we supply a lower bound for the
number of equivalence classes of real rational functions for each given
ramification profile. Furthermore, as a consequence, we obtain a new proof for
a theorem ( 10.2307/3062151 , 10.4007/annals.2009.170.863 ,
10.1090/S0894-0347-09-00640-7 ) that corresponds to a special case of a reality
problem in enumerative geometry which was known as the B. \& M. Shapiro
Conjecture, now it is a theorem \cite{MR2552110}. The theorem version that we
prove concerns generic rational functions, assuring that if all critical points
of that function are real, then we can transform it into a rational map with
real coefficients by post-composition with an automorphism of
$\mathbb{C}\mathbb{P}^{1}$. The proof we present is constructive and founded on
elementary arguments.}
