Walks on SPR Neighborhoods

A nearest-neighbor-interchange (NNI)-walk is a sequence of unrooted phylogenetic trees, $(T_1, T_2, \ldots, T_k)$ where each consecutive pair of trees differs by a single NNI move. We give tight bounds on the length of the shortest NNI-walks that visit all trees in a subtree-prune-and-regraft (SPR) neighborhood of a given tree. For any unrooted, binary tree, $(T)$, on $(n)$ leaves, the shortest walk takes $(\Theta (n^2))$ additional steps more than the number of trees in the SPR neighborhood. This answers Bryant's Second Combinatorial Challenge from the Phylogenetics Challenges List, the Isaac Newton Institute, 2011, and the Penny Ante Problem List, 2009.