Walks on SPR Neighborhoods

A nearest-neighbor-interchange (NNI)-walk is a sequence of unrooted phylogenetic trees, T₁,T₂, ... ,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 ⊖(n²) 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.