Unavoidable subtrees

Let Tk be a family of all k-vertex trees. For T⊆Tk and a tree T, we write T→T if T contains at least one of the trees from T as a subtree, we write T⁄→T otherwise. Let ex(T) be the smallest integer n, if such exists, such that for any tree T on at least n vertices T→T. It is shown that min{ex(T):T⊆Tk,|T|=q}=2Θ(klogq−1k), where logq−1 is the q−1 times iterated logarithm. In addition, the bounds on ex(T) for families T with a given number of spiders are given.