A Ramsey type result for oriented trees

Given positive integers h and k, denote by r(h,k) the smallest integer n such that in any k-coloring of the edges of a tournament on more than n vertices there is a monochromatic copy of every oriented tree on h vertices. We prove that r(h,k)=(h−1)^k for all k sufficiently large (k=Θ(hlogh) suffices). The bound (h−1)^k is tight. The related parameter r^∗(h,k) where some color contains all oriented trees is asymptotically determined. Values of r(h,2) for some small h are also established.