Combinatorial description of a moduli space of curves and of extremal polynomials

For the description of extremal polynomials (that is, the typical solutions of least deviation problems) one uses real hyperelliptic curves. A partitioning of the moduli space of such curves into cells enumerated by trees is considered. As an application of these techniques the range of the period map of the universal cover of the moduli space is explicitly calculated. In addition, extremal polynomials are enumerated by weighted graphs.