We investigate a problem in extremal graph theory known as the Erd?s-S?s conjecture. In 2 2 1959 Erd?s and Gallai proved every graph G of order n with e ( G ) n ( k ? 1) contains a path with k edges. Motivated by the result, Erd?s and S?s made the following conjecture in 1963. Erd?s-S?s Conjecture ( E S C ): If G is a graph of order n with e ( G ) n ( k ? 1) , then G contains every tree T with k edges as a subgraph. Note that e ( G ) n ( k ? 1) if and only if e ( G ¯) n ( n ? k ) . Thus the Erd?s-S?s conjecture can 2 2 be restated as follows. 2 Suppose that G is a graph of order n and T is any tree of size k . If e ( G ) n ( n ? k ) , then G ¯ contains T as a subgraph. However, approximate versions of the conjecture were proved by Ajtai, koml?s and Sze- merédi using the Regularity Lemma. They showed that for su?ciently large k and su?ciently large n ESC is true for all graphs of order n that satisfy the hypothesis. This conjecture is still open. There are some partial results, mainly on two directions on 18 this conjecture. One is to pose conditions on the graph G . In this thesis, we consider the problem of the Erd?s-S?s conjecture for the following cases. • Tree T has diameter at most 5. • G contains no K 2 ,s as a subgraph.