  Let $G$ be a finite simple graph and let $I(G)$ be its edge ideal. In this
article, we deeply investigate the squarefree powers of $I(G)$ by means of
Betti splittings. When $G$ is a forest, it is shown that the normalized depth
function of $I(G)$ is non-increasing. Furthermore, we compute explicitly the
regularity function of squarefree powers of $I(G)$ with $G$ a forest,
confirming a conjecture of Erey and Hibi.
