In this paper, we have proposed altogether different and new approaches for the computation of all possible spanning trees of a simple, undirected, and connected graph. Our proposed algorithms have the capability to solve the major bottlenecks in this area, namely, generation of duplicate trees and circuit checking. In the first algorithm, the given graph has been converted to its corresponding weighted complete graph, which proposes to have novelties and limitations of its own. In addition, we have also proposed another related algorithm, and as a result, we have been able to come up with new ideas in this research domain of graph theory. © 2013 IEEE.