We have developed a phylogenetic tree reconstruction method that detects and reports multiple, topologically distant, low cost solutions. Our method is a generalization of the Neighbor-Joining (NJ) method of Nei and Saitou, and affords a more thorough sampling of the solution space by keeping track of multiple partial solutions during its execution. The scope of the solution space sampling is controlled by a pair of user-specified parameters - the total number of alternate solutions and the number of alternate solutions that are randomly selected - effecting a smooth tradeoff between run time and solution quality and diversity. This method can discover topologically distinct low cost solutions. In tests on biological and synthetic datasets using either the least-squares distance or minimum-evolution criterion, the method consistently performed as well as, or better than, either the Neighbor-Joining heuristic or the PHYLIP implementation of the Fitch-Margoliash distance measure. In addi..