Let $T$ be an arbitrary phylogenetic tree with $n$ leaves. It is well-known that the average quartet distance between two assignments of taxa to the leaves of $T$ is $\frac 23 \binom{n}{4}$. However, a longstanding conjecture of Bandelt and Dress asserts that $(\frac 23 +o(1))\binom{n}{4}$ is also the {\em maximum} quartet distance between two assignments. While Alon, Naves, and Sudakov have shown this indeed holds for caterpillar trees, the general case of the conjecture is still unresolved. A natural extension is when partial information is given: the two assignments are known to coincide on a given subset of taxa. The partial information setting is biologically relevant as the location of some taxa (species) in the phylogenetic tree may be known, and for other taxa it might not be known. What can we then say about the average and maximum quartet distance in this more general setting? Surprisingly, even determining the {\em average} quartet distance becomes a nontrivial task in the partial information setting and determining the maximum quartet distance is even more challenging, as these turn out to be dependent of the structure of $T$. In this paper we prove nontrivial asymptotic bounds that are sometimes tight for the average quartet distance in the partial information setting. We also show that the Bandelt and Dress conjecture does not generally hold under the partial information setting. Specifically, we prove that there are cases where the average and maximum quartet distance substantially differ.

翻译：以美元计叶。 众所周知, 两组分级之间的四重奏平均距离是 $23\ binom{ n ⁇ 4}美元。 然而, 班德尔特和德雷斯的长期猜想显示, $23+1)\ binom{ n ⁇ 4} 美元也是四重奏之间的最大距离。 Alon, Naves和Sudakov 显示, 这确实保持了毛球树的距离, 猜想中的一般情况仍然没有解决。 当给出部分信息时, 自然延伸是 : 这两项分级在给定的分级上是已知的。 部分信息设置在生物上具有相关性, 某些分级( 物种) 在血色树中可能已经知道, 而其他分级则可能不知道。 我们的中位和最高四重奏在这种总体情况下, 我们的平均和最高距离是多少? 令人惊讶的是, 即便在四重音乐中, 最远的信息也证明, 最远的分界结构中, 也显示我们平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平。 。