## Abstract

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 quartet distance between the two trees is \(\frac{2}{3}\left( {\begin{array}{c}N\\ 4\end{array}}\right) \). 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* bits long 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 \(\left( \frac{2}{3} + o(1)\right) \left( {\begin{array}{c}N\\ 4\end{array}}\right) \), and it always exceeds the \(\frac{2}{3}\left( {\begin{array}{c}N\\ 4\end{array}}\right) \) bound.

## Keywords

Phylogenetic trees Quartet distance Prefix and suffix orders## Mathematics Subject Classification

05C05 68R15 92D15## Notes

### Acknowledgements

Thanks to Stefan Grünewald for helpful discussions, which motivated this work. We would also like to thank Noga Alon and Gil Cohen for their help regarding what “explicit construction” exactly means. PLE was supported in part by the Hungarian NSF Grant K116769. Part of this work was done when PLE visited BC, supported by an exchange program of the Hungarian and Israeli Academies of Sciences. BC was supported by a Grant from the Blavatnik Computer Science Research Fund, and by the LTZI (Long Term Zero Income) fund of the ISF (Israeli Science Foundation).

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