In this thesis, we study the ramsey goodness of paths and bounded degree trees. Given a pair of graphs G and H, the Ramsey number R(G,H) is defined to be the smallest positive integer n such that every ?-coloring of the edges of the complete graph Kn with colors red and blue contains a red copy of G or a blue copy of H. Frank Ramsey proved these numbers are always finite, therefore R(G,H) is a meaningful definition. Determining or computing the Ramsey number is one of the most important and difficult problems in Combinatorics. In this thesis, it is shown that we can compute R(G,H) with a simple formula for graphs G and H with special properties. Let G is a connected graph and H is a graph with chromatic number #??;X(H) (the smallest number of colors needed to color the vertices of H so that no pair of adjacent vertices have the same color) and the parameter X #??;(H) (the size of the smallest color class between all of #??;X(H)-coloring of H).
