Computing the edge betweenness centrality is an important step in a great deal of the analysis tasks of community structures in complex networks. It mostly serves as a measure for the traffic or flow of a particular edge in connecting various parts or communities together. Various algorithms that compute the edge betweenness centrality in general graphs exist but they are expensive. In this paper, we design an algorithm that takes advantage of the structure of tree graphs to compute the edge betweenness centrality more efficiently in such graphs and perform experiments on random graphs.