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A classical result of Robertson and Seymour states that the set of graphscontaining a fixed planar graph $H$ as a minor has the so-calledErd\H{o}s-P\'osa property; namely, there exists a function $f$ depending onlyon $H$ such that, for every graph $G$ and every positive integer $k$, the graph$G$ has $k$ vertex-disjoint subgraphs each containing $H$ as a minor, or thereexists a subset $X$ of vertices of $G$ with $|X| \leq f(k)$ such that $G - X$has no $H$-minor. While the best function $f$ currently known is exponential in$k$, a $O(k \log k)$ bound is known in the special case where $H$ is a forest.This is a consequence of a theorem of Bienstock, Robertson, Seymour, and Thomason the pathwidth of graphs with an excluded forest-minor. In this paper we showthat the function $f$ can be taken to be linear when $H$ is a forest. This isbest possible in the sense that no linear bound is possible if $H$ has a cycle.

Excluded Forest Minors and the Erd\H{o}s-P\'osa Property