International Journal of Mathematical Research 2311-7427 2306-2223 10.18488/journal.24/2015.4.2/24.2.76.100 International Journal of Mathematical Research On the Component Analysis and Transformation of An Explicit Fifth – Stage Fourth – Order Runge – Kutta Methods International Journal of Mathematical Research International Journal of Mathematical Research 04-2015 2015 04-2015 04-2015 4 2 76 100 30 Nov 30 Nov This work is designed to transform the fifth stage – fourth order explicit Runge-Kutta method with the aim of projecting a new method of implementing it through tree diagram analysis. Efforts will be made to represent the equations derived from the y derivatives and x,y derivatives separately on Butcher’s rooted trees. This is because the rooted trees and derived equations for the y derivatives and x,y derivatives are the same for the explicit fourth-stage fourth-order methods, hence, we are motivated to analyze the fifth-stage fourth-order method. This idea is also derivable from general graphs and combinatorics.