Fast-converging methods for reconstructing phylogenetic trees require that the sequences characterizing the taxa be of only polynomial length, a major asset in practice, since real-life sequences are of bounded length. However, of the half-dozen such methods proposed over the last few years, only two fulfill this condition without requiring knowledge of typically unknown parameters, such as the evolutionary rate(s) used in the model; this additional requirement severely limits the applicability of the methods. We say that methods that need such knowledge demonstrate relative fast convergence, since they rely upon an oracle. We focus on the class of methods that do not require such knowledge and thus demonstrate absolute fast convergence. We give a very general construction scheme that not only turns any relative fast-converging method into an absolute fast-converging one, but also turns any statistically consistent method that converges from sequence of length &Ogr;(e &Ogr;(diam(T))) into an absolute fast-converging method.