Fig. 1

Illustration of the NNI operation on phylogenetic trees. A In a phylogenetic tree, an NNI operation on an internal edge (a, b) first selects two edges (a, x) and (b, y) that are, respectively, incident to a and b such that \((a, x)\ne (a, b)\ne (y, b)\); it then rewires them to the opposite end so that (a, y) and (b, x) are the two edges in the resulting tree (red). Since a and b are labeled differently, a unrooted tree can be transformed into one of four possible trees in one NNI. B In a rooted phylogenetic tree T, an NNI operation on an internal edge (a, b) (where b is a child of a) transforms T by either (i) selecting two edges (a, x) and (b, y) that leave from a and b, respectively, and replacing them with (a, y) and (b, x) (left), where \(x\ne b\), or (ii) selecting an edge (b, y) leaving from b and replacing the unique edge (z, a) that enters a, (a, b) and (b, y) with (z, b), (b, a) and (a, y) (right), respectively. A rooted tree can be transformed into four different trees in one NNI. C An illustration of the interchange of two labels of the ends of an internal edge in two NNIs in an 1-labeled phylogenetic tree