Tree densities in sparse graph classes

What is the maximum number of copies of a fixed forest T in an n-vertex graph in a graph class as ? We answer this question for a variety of sparse graph classes. In particular, we show that the answer is where is the size of the largest stable set in the subforest of T induced by the vertices of degree at most d, for some integer d that depends on. For example, when is the class of k-degenerate graphs then; when is the class of graphs containing no -minor then; and when is the class of k-planar graphs then. All these results are in fact consequences of a single lemma in terms of a finite set of excluded subgraphs.