On the Erdos-Sos Conjecture and graphs with large minimum degree

Suppose G is a simple graph with average vertex degree greater than k - 2. Erdos and Sos conjectured that G contains every tree on k vertices. Sidorenko proved G contains every tree that has a vertex v with at least inverted right perpendiculark/2inverted left perpendicular - 1 leaf neighbors. We prove this is true if v has only inverted right perpendiculark/2inverted left perpendicular - 2 leaf neighbors. We generalize Sidorenko's result by proving that if G has minimum degree d, then G contains every tree that has a vertex with least (k - 1) - d leaf neighbors. We use these results to prove that if G has average degree greater than k - 2 and minimum degree at least k - 4, then G contains every tree on k vertices.