Many phylogenetic algorithms search the space of possible trees using topological rearrangements and some optimality criterion. FastME is such an approach that uses the {em balanced minimum evolution (BME)} principle, which computer studies have demonstrated to have high accuracy. FastME includes two variants: {em balanced subtree prune and regraft (BSPR)} and {em balanced nearest neighbor interchange (BNNI)}. These algorithms take as input a distance matrix and a putative phylogenetic tree. The tree is modified using SPR or NNI operations, respectively, to reduce the BME length relative to the distance matrix, until a tree with (locally) shortest BME length is found. Following computer simulations, it has been conjectured that BSPR and BNNI are consistent, i.e. for an input distance that is a tree-metric, they converge to the corresponding tree. We prove that the BSPR algorithm is consistent. Moreover, even if the input contains small errors relative to a tree-metric, we show that the BSPR algorithm still returns the corresponding tree. Whether BNNI is consistent remains open.