Recently we developed a new quartet-based algorithm for phylogenetic analysis [22]. This algorithm constructs a limited number of trees for a given set of DNA or protein sequences and the initial experimental results show that the probability for the correct tree to be included in this small set of trees is very high. In this paper we further extend the idea. We first discuss a revision to the original algorithm to reduce the number of trees generated, while keeping the high probability for the correct tree to be included. We then deal with the issue on how to retrieve the correct tree from the generated trees and our current approach is to calculate the likelihood values of these trees and pick up a few best ones which have the highest likelihood values. Though the experimental results are comparable to that obtained from currently popular ML based algorithms, we find that it is common that certain incorrect trees can have likelihood values at least as large as that of the correct tree. A significant implication of this is that even if we are able to find a truly globally optimal tree under the maximum likelihood criterion, this tree may not necessarily be the correct phylogenetic tree!
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