  A nearest-neighbor-interchange (NNI) walk is a sequence of unrooted
phylogenetic trees, T_0, T_1, T_2,... where each consecutive pair of trees
differ by a single NNI move. We give tight bounds on the length of the shortest
NNI-walks that visit all trees in an 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 than the number of trees
in the SPR neighborhood. This answers Bryant's Second Combinatorial Conjecture
from the Phylogenetics Challenges List, the Isaac Newton Institute, 2011, and
the Penny Ante Problem List, 2009.
