Universality for bounded degree spanning trees in randomly perturbed graphs

We solve a problem of Krivelevich, Kwan and Sudakov concerning the threshold for the containment of all bounded degree spanning trees in the model of randomly perturbed dense graphs. More precisely, we show that, if we start with a dense graph G(alpha) on n vertices with delta(G(alpha)) >= alpha n for alpha > 0 and we add to it the binomial random graph G(n,C/n), then with high probability the graph G(alpha)?G(n,C/n) contains copies of all spanning trees with maximum degree at most Delta simultaneously, where C depends only on alpha and Delta.