Application of betti splittings to the regularity of binomial edge ideals

Let [Formula: see text] be a simple graph on [Formula: see text] vertices and [Formula: see text] denote the corresponding binomial edge ideal in [Formula: see text], where [Formula: see text] is a field. In this paper, we provide an upper bound for the regularity of binomial edge ideals of trees. Our approach leverages the recently developed framework of Betti splittings of binomial edge ideals, a powerful technique for studying homological invariants via decompositions into simpler ideals. We also give a lower bound for the same. As a consequence, we explicitly compute the regularity of binomial edge ideals of trees under certain conditions.