We consider the problem of finding generators of the toric ideal associated to a
combinatorial object called a staged tree. Our motivation to consider this problem
originates from the use of staged trees to represent discrete statistical models such as
conditional independence models and discrete Bayesian networks. The main theorem
in this article states that toric ideals of staged trees that are balanced and stratified
are generated by a quadratic Gröbner basis whose initial ideal is square-free. We
apply this theorem to construct Gröbner bases of a subclass of discrete statistical
models represented by staged trees. The proof of the main result is based
on Sullivant’s toric fiber product construction (J. Algebra 316:2 (2007),
560–577).