Neighborhoods of trees in circular orderings

In phylogenetics, a common strategy used to construct an evolutionary tree for a set of species [Formula: see text] is to search in the space of all such trees for one that optimizes some given score function (such as the minimum evolution, parsimony or likelihood score). As this can be computationally intensive, it was recently proposed to restrict such searches to the set of all those trees that are compatible with some circular ordering of the set [Formula: see text]. To inform the design of efficient algorithms to perform such searches, it is therefore of interest to find bounds for the number of trees compatible with a fixed ordering in the neighborhood of a tree that is determined by certain tree operations commonly used to search for trees: the nearest neighbor interchange (NNI), the subtree prune and regraft (SPR) and the tree bisection and reconnection (TBR) operations. We show that the size of such a neighborhood of a binary tree associated with the NNI operation is independent of the tree's topology, but that this is not the case for the SPR and TBR operations. We also give tight upper and lower bounds for the size of the neighborhood of a binary tree for the SPR and TBR operations and characterize those trees for which these bounds are attained.