ERRATUM TO “ON THE EDGE l∞ RADIUS OF SAITOU AND NEI’S METHOD FOR PHYLOGENETIC RECONSTRUCTION” [THEORET

Theorem 2 of [2], which claims to settle Atteson’s edge radius conjecture [1], is invalidly proven. The argument in [2] is inductive, and is based on the assumption that if initially ||D̂ − DT ||∞ < l(e) 4 then as the algorithm proceeds, the intermediate distance matrices obtained by agglomeration are within l∞ distance l(e) 4 from some tree metric (this point is in fact not explicitly stated in [2]). Such a result holds in the case where one is proving the standard radius bound (Lemma 7 in [1]), since in that case there is a guarantee of only collapsing pairs of nodes forming a cherry in the correct model tree. However, the analysis fails for the edge radius argument because the agglomerated leaves at any step may not form a cherry. In this case, it is not at all obvious how to find a reduced model topology T ′ that is consistent in some weak sense with the initial model tree and that allows the continuation of the induction argument by satisfying ||D̂′ − DT ′||∞ < l(e) 4 (where D̂′ is the result of the collapsing step on on the distance matrix D′). In fact, in Theorem 34 in [3] we provide an example in which such a tree does not exist. Our example shows that the intermediate distance matrices may be further than l(e) 4 from any tree metric. This presents a problem not only for the proof of Theorem 2 of [2], but also for Theorem 4 of [4]. The edge radius theorem can be proved inductively by relaxing the hypothesis, as is done in Theorem 25 of [3].