  Given an integer $g \geq 0$ and a weight vector $w \in \mathbb{Q}^n \cap (0,
1]^n$ satisfying $2g - 2 + \sum w_i > 0$, let $\Delta_{g, w}$ denote the moduli
space of $n$-marked, $w$-stable tropical curves of genus $g$ and volume one. We
calculate the automorphism group $\mathrm{Aut}(\Delta_{g, w})$ for $g \geq 1$
and arbitrary $w$, and we calculate the group $\mathrm{Aut}(\Delta_{0, w})$
when $w$ is heavy/light. In both of these cases, we show that
$\mathrm{Aut}(\Delta_{g, w}) \cong \mathrm{Aut}(K_w)$, where $K_w$ is the
abstract simplicial complex on $\{1, \ldots, n\}$ whose faces are subsets with
$w$-weight at most $1$. We show that these groups are precisely the finite
direct products of symmetric groups. The space $\Delta_{g, w}$ may also be
identified with the dual complex of the divisor of singular curves in the
algebraic Hassett space $\overline{\mathcal{M}}_{g, w}$. Following the work of
Massarenti and Mella on the biregular automorphism group
$\mathrm{Aut}(\overline{\mathcal{M}}_{g, w})$, we show that
$\mathrm{Aut}(\Delta_{g, w})$ is naturally identified with the subgroup of
automorphisms which preserve the divisor of singular curves.
