Depth and regularity of powers of edge ideals of edge-weighted trees

For an increasing weighted tree $G_{\omega }$, we obtain an asymptotic value and a sharp bound on the index stability of the depth function of its edge ideal $I(G_{\omega })$. Moreover, if $G_{\omega }$ is a strictly increasing weighted tree, we provide the minimal free resolution of $I(G_{\omega })$ and an exact formula for the regularity of all powers of $I(G_{\omega })$.