On the Neighborhoods of Trees

Tree rearrangement operations typically induce a metric on the space of phylogenetic trees. One important property of these metrics is the size of the neighborhood, that is, the number of trees exactly one operation from a given tree. We present an exact expression for the size of the TBR (tree bisection and reconnection) neighborhood, thus answering a question first posed by Allen and Steel . In addition, we also obtain a characterization of the extremal trees whose TBR neighborhoods are maximized and minimized.