  In this paper we study the moduli space of the tropicalizations of Riemann
surfaces. We first tropicalize a smooth pointed Riemann surface by a graph
defined by its (hyperbolic) pair of pants decomposition. Then we can construct
the moduli space of tropicalizations based on a fixed regular tropicalization,
and compactify it by adding strata parametrizing weighted contractions. We show
that this compact moduli space is also Hausdorff. In the end, we compare this
moduli space with the moduli space of Riemann surfaces, establishing a partial
order-preserving correspondence between the stratifications of these two moduli
spaces.
