Given a compact Riemann surface $X$, there exists a bijective correspondence between Belyi's functions (namely, coverings $f: X \to \mathbb{C}_\infty$ of $\mathbb{C}_\infty$ with exactly three branch points) and Grothendieck's dessins d'enfants on $X$. Let $X=\mathbb{C}_\infty$. In such a case, we may consider coverings fixing $\infty$. Hence, the resulting dessins are trees and the corresponding Belyi's functions are called Shabat polynomials, since they are actually polynomials. Now, my questions are:
1) Given a Shabat polynomial, how may I represent the corresponding plane tree?
2) Given a tree in the plane, how may I derive the corresponding Shabat polynomial?
3) More in general, assume that a given bipartite plane graph is not a tree. When it is a dessin d'enfant, how may I derive the corresponding Belyis' function? Conversely, given a Belyi's function $f: \mathbb{C}_\infty \to \mathbb{C}_\infty$, how may I represent the corresponding Grothendieck's dessin?
May you also suggest a complete list of references (books, papers and free computer algebra systems) on the problem of computing Belyi's functions starting from a dessin? For example, the classical Girondo's textbook is very useful but has no the answers I'm finding for.
1) Your shabat polynomial has two critical values, that can be normed to be 0 and 1. Now you get your tree by drawing $p^{-1}([0,1])$ in the complex plane.
2) This is a very hard problem, and in general there is no way known to do that (Belyi's theorem is an existence theorem, its proof does not give an explicit construction), except solving very large systems of equations.
3) Again, only very large systems can give you the solution. In the other way, you get the dessin by drawing p^{-1}([0,1]).