  We study the locus of tropical hyperelliptic curves inside the moduli space
of tropical curves of genus g. We define a harmonic morphism of metric graphs
and prove that a metric graph is hyperelliptic if and only if it admits a
harmonic morphism of degree 2 to a metric tree. This generalizes the work of
Baker and Norine on combinatorial graphs to the metric case. We then prove that
the locus of 2-edge-connected genus g tropical hyperelliptic curves is a
(2g-1)-dimensional stacky polyhedral fan whose maximal cells are in bijection
with trees on g-1 vertices with maximum valence 3. Finally, we show that the
Berkovich skeleton of a classical hyperelliptic plane curve satisfying a
certain tropical smoothness condition lies in a maximal cell of genus g called
a standard ladder.
