ENJ algorithm can construct triple phylogenetic trees.

Phylogenetic analysis is used to analyze the evolution of species according to the characteristics of biological sequences. The analytical results are generally represented by phylogenetic trees. NJ (neighbor joining) is a frequently used algorithm for constructing phylogenetic trees because of its few assumptions, fast operation, and high accuracy, and is based on the distance between taxa. It is known that NJ usually constructs different phylogenetic trees for the same dataset with differences in input order, which are known as "tied trees." This article proposes an improved method of NJ, called ENJ (extended neighbor joining). The ENJ can join several (currently limited to three) nodes with the same minimum distance into a new node, rather than joining two nodes in one iteration, so it can construct triple phylogenetic trees. We have inferred the formulas for updating the distance values and calculating the branch lengths for the ENJ algorithm. We have tested the ENJ with simulated and real data. The experimental results show that, compared with other methods, the trees constructed by the ENJ have greater similarity to the initial trees, and the ENJ is much faster than the NJ algorithm. Moreover, we have constructed a phylogenetic tree for the novel coronavirus (COVID-19) and related coronaviruses by ENJ, which shows that COVID-19 and SARS-CoV are closer than other coronaviruses. Because it differs from the existing phylogenetic trees for those coronaviruses, we constructed a phylogenetic network for them. The network shows those species have had a reticulate evolution.