Studies are carried out on the uniqueness of the stationary point on the likelihood function for estimating molecular phylogenetic trees, yielding proof that there exists at most one stationary point, i.e., the maximum point, in the parameter range for the one parameter model of nucleotide substitution. The proof is simple yet applicable to any type of tree topology with an arbitrary number of operational taxonomic units (OTUs). The proof ensures that any valid approximation algorithm be able to reach the unique maximum point under the conditions mentioned above. An algorithm developed incorporating Newton's approximation method is then compared with the conventional one by means of computer simulation. The results show that the newly developed algorithm always requires less CPU time than the conventional one, whereas both algorithms lead to identical molecular phylogenetic trees in accordance with the proof.