I have the following question about the paper A parsimony-based metric for phylogenetic trees by V. Moulton and T. Wu.
Lemma 6.2 states that only caterpillar trees have unit distance to some other tree. This seems to suggest that for any non-caterpillar tree, the 1-neighborhood should be empty. It is also shown in the paper that the parsimony-distance neighborhood contains the TBR-distance neighborhood (where TBR = tree bisection and reconnection). However, the TBR-neighborhood of any tree should be non-empty, since it is possible to apply a TBR move and obtain a different tree. Are there some assumptions I am missing about when the parsimony-neighborhood is empty, or when the TBR-neighborhood is contained in the parsimony neighborhood?
Also, when applying the formula in Corollary 6.6 to the non-caterpillar tree on 6 leaves, the formula seems to give the neighborhood size $|N_p| = 54$, while Lemma 6.2 suggests that the neighborhood is empty. Should applying this formula really give $|N_p| = 0$, which I am misreading somehow?