#### Abstract

A conjecture of Bandelt and Dress states that the maximum quartet distance between any two phylogenetic trees on n leaves is at most ( 23 + o(1)) ( n 4 ) . Using the machinery of flag algebras we improve the currently known bounds regarding this conjecture, in particular we show that the maximum is at most (0.69 + o(1)) ( n 4 ) . We also give further… (More)