The problem of improving the computational efficiency of a method for an iterative solution of a linear circuit equation, known as the tree relaxation (TR) method, is investigated. It is shown that the adoption of the tree-branch voltages instead of the node voltages, resulting in a tree relaxation modified (TR) method, represents a more advantageous choice for the network variables. The extra processing needed for setting up the fundamental cut-set matrix allows a significant reduction of the computational burdens to be obtained. These savings are even more substantial if a transient analysis is to be performed. It is shown that the choice of the tree-branch voltages yields further advantages if a parallel version of the TRM is to be implemented.>