Embedding spanning trees in random graphs

We prove that if T is a tree on n vertices with maximum degree Δ and the edge probability p(n) satisfies np ≥ C max{Δ log n, n ^∈} for some constant ∈ > 0, then with high probability the random graph G(n, p) contains a copy of T. The obtained bound on the edge probability is shown to be essentially tight for Δ = n ^Θ(1).