  A staged tree model is a discrete statistical model encoding relationships
between events. These models are realised by directed trees with coloured
vertices. In algebro-geometric terms, the model consists of points inside a
toric variety. For certain trees, called balanced, the model is in fact the
intersection of the toric variety and the probability simplex. This gives the
model a straightforward description, and has computational advantages. In this
paper we show that the class of staged tree models with a toric structure
extends far outside of the balanced case, if we allow a change of coordinates.
It is an open problem whether all staged tree models have toric structure.
