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The nef cone of a projective toric variety is a polyhedral cone associated to the polytope or polyhedral fan defining the toric variety. I will describe three combinatorially defined polyhedral cones associated to a polyhedral fan, which all equal the nef cone when the fan is the normal fan of a polytope, and introduce the question of understanding when these are equal. This is particularly interesting when the polyhedral fan is the space of phylogenetic trees, when the cones bound the nef cone of moduli space of genus zero curves with n marked points. This is joint work with Angela Gibney (Georgia).