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We solve a problem of Krivelevich, Kwan and Sudakov [SIAM Journal on DiscreteMathematics 31 (2017), 155-171] concerning the threshold for the containment ofall bounded degree spanning trees in the model of randomly perturbed densegraphs. More precisely, we show that, if we start with a dense graph $G_\alpha$on $n$ vertices with $\delta(G_\alpha)\ge \alpha n$ for $\alpha>0$ and we addto it the binomial random graph $G(n,C/n)$, then with high probability thegraph $G_\alpha\cup G(n,C/n)$ contains copies of all spanning trees withmaximum degree at most $\Delta$ simultaneously, where $C$ depends only on$\alpha$ and $\Delta$.

Universality for bounded degree spanning trees in randomly perturbed graphs