The Erdős-Sós conjecture for graphs of girth 5

We prove that every graph of girth at least 5 with minimum degree δ ⩾ k/2 contains every tree with k edges, whose maximum degree does not exceed the maximum degree of the graph. An immediate consequence is that the famous Erdős-Sós Conjecture, saying that every graph of order n with more than n(k − 1)/2 edges contains every tree with k edges, is true for graphs of girth at least 5.