Tight bound for the Erdős-Pósa property of tree minors

Let T be a tree on t vertices. We prove that for every positive integer k and every graph G, either G contains k pairwise vertex-disjoint subgraphs each having a T minor, or there exists a set X of at most t(k-1) vertices of G such that G-X has no T minor. The bound on the size of X is best possible and improves on an earlier f(t)k bound proved by Fiorini, Joret, and Wood (2013) with some very fast growing function f(t). Moreover, our proof is very short and simple.