Given an integer g 0 and a weight vector w 2 Q n \ .0; 1 n satisfying 2g 2 C P w i > 0, let g;w denote the moduli space of n-marked, w-stable tropical curves of genus g and volume one. We calculate the automorphism group Aut. g;w / for g 1 and arbitrary w, and we calculate the group Aut. 0;w / when w is heavy/light. In both of these cases, we show that Aut. g;w / Š Aut.K w /, where K w is the abstract simplicial complex on ¹1; : : : ; nº 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 g;w may also be identified with the dual complex of the divisor of singular curves in the algebraic Hassett space M g;w . Following the work of Massarenti and Mella (2017) on the biregular automorphism group Aut.M g;w /, we show that Aut. g;w / is naturally identified with the subgroup of automorphisms which preserve the divisor of singular curves.

Given an integer g ≥ 0 and a weight vector w ∈ Q n ∩(0, 1] n satisfying 2g−2+ wi > 0, let ∆g,w denote the moduli space of n-marked, w-stable tropical curves of genus g and volume one. We calculate the automorphism group Aut(∆g,w) for g ≥ 1 and arbitrary w, and we calculate the group Aut(∆0,w) when w is heavy/light. In both of these cases, we show that Aut(∆g,w) ∼ = Aut(Kw), where Kw is the abstract simplicial complex on {1, . . . , 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 ∆g,w may also be identified with the dual complex of the divisor of singular curves in the algebraic Hassett space Mg,w. Following the work of Massarenti and Mella [MM17] on the biregular automorphism group Aut(Mg,w), we show that Aut(∆g,w) is naturally identified with the subgroup of automorphisms which preserve the divisor of singular curves. Contents1. Introduction 1 2. Graphs and ∆ g,w 5 3. Calculation of Aut(∆ g,w ) for g ≥ 1 8 4. The genus 0 case 14 Appendix A.