We study the convergence rates of neighbor-joining and several new phylogenetic reconstruction methods on families of trees of bounded diameter. Our study presents theoretically obtained convergence rates, as well as an empirical study based upon simulation of evolution on random birth-death trees. We find that the new phylogenetic methods offer an advantage over the neighbor-joining method, except at low rates of evolution where they have comparable performance. The improvement in performance of the new methods over neighbor-joining increases with the number of taxa and the rate of evolution.
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