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A High Quartets Distance Construction

Given two binary trees on $N$ labeled leaves, the quartet distance between the trees is the number of disagreeing quartets. By permuting the leaves at random, the expected quartets distance between the two trees is $\frac{2}{3}\binom{N}{4}$. However, no strongly explicit construction reaching this bound asymptotically was known. We consider complete, balanced binary trees on $N=2^n$ leaves, labeled by $n$ long bit sequences. Ordering the leaves in one tree by the prefix order, and in the other tree by the suffix order, we show that the resulting quartet distance is $\left(\frac{2}{3} + o(1)\right)\binom{N}{4}$, and it always exceeds the $\frac{2}{3}\binom{N}{4}$ bound.