  We study moduli spaces of rational weighted stable tropical curves, and their
connections with the classical Hassett spaces. Given a vector w of weights, the
moduli space of tropical w-stable curves can be given the structure of a
balanced fan if and only if w has only heavy and light entries. In this case,
we can express the moduli space as the Bergman fan of a graphic matroid.
Furthermore, we realize the tropical moduli space as a geometric
tropicalization, and as a Berkovich skeleton, of the classical moduli space.
This builds on previous work of Tevelev, Gibney--Maclagan, and
Abramovich--Caporaso--Payne. Finally, we construct the moduli spaces of
heavy/light weighted tropical curves as fiber products of unweighted spaces,
and explore parallels with the algebraic world.
