We have recently shown that balanced minimum evolution (BME), the principle underlying neighbour-joining, has optimal robustness, whereas another version of minimum evolution based on least squares has very limited robustness. This difference may partly explain the well-documented gap in reconstruction accuracy between these two approaches. Here, I will announce a result that considerably strengthens the result above: I show that BME is in fact the only principle with optimal robustness, among all linear optimization principles, i.e., those that score tree topologies on the basis of linear functions of the input distances. This includes all minimum evolution principles in the line initiated by Rzhetsky and Nei (1992).