Quickly excluding an apex-forest

We give a short proof that for every apex-forest X on at least two vertices, graphs excluding X as a minor have layered pathwidth at most 2|V(X)|-3. This improves upon a result by Dujmović, Eppstein, Joret, Morin, and Wood (SIDMA, 2020). Our main tool is a structural result about graphs excluding a forest as a rooted minor, which is of independent interest. We develop similar tools for treedepth and treewidth. We discuss implications for Erdős-Pósa properties of rooted models of minors in graphs.