Given a statistically consistent quartet estimator, a family of statistically consistent tree inference algorithms can be designed (Additional file 1: Algorithm S1). Details and proofs are given in the (Additional file 1: Section 2.4). The basic idea is the following. We can use the quartet estimator to infer a distance matrix that asymptotically can be made arbitrarily close to an additive distance matrix for the true tree topology. Using a method such as neighbor-joining that infers the correct tree for additive distance matrices with a safety radius will give a consistent estimator of the tree .