“Lassoing”一个系统发育树I：基本属性、外壳和覆盖
`Lassoing' a phylogenetic tree I: Basic properties, shellings, and covers
摘要：
一个经典的结果，进化生物学的基础，指出边加权树$T$的叶集$X$，积极的边缘权重，没有2度的顶点可以唯一地重建从一组叶到叶之间的距离$X$的任何两个元素。在生物学中，$X$对应于一组分类群（例如现存物种），树$T$描述了它们的系统发育关系，边缘对应于较早的物种进化一段时间，直到通过一些物种形成/分叉事件分裂成两个或多个物种，它们的长度对应于在这样的物种中随着时间的推移积累的遗传变化。在本文中，我们调查哪些子集的$\binom{X}{2}$足以确定（“套索”）一棵树从叶到叶的距离引起的树。这个问题特别具有话题性，因为对遗传距离的可靠估计--即使（如果不是特别的话）通过现代大规模测序方法--一般来说也只能用于某些分类群的组合。
英文摘要：
A classical result, fundamental to evolutionary biology, states that an edge-weighted tree $T$ with leaf set $X$, positive edge weights, and no vertices of degree 2 can be uniquely reconstructed from the set of leaf-to-leaf distances between any two elements of $X$. In biology, $X$ corresponds to a set of taxa (e.g. extant species), the tree $T$ describes their phylogenetic relationships, the edges correspond to earlier species evolving for a time until splitting in two or more species by some speciation/bifurcation event, and their length corresponds to the genetic change accumulating over that time in such a species. In this paper, we investigate which subsets of $\binom{X}{2}$ suffice to determine (`lasso') a tree from the leaf-to-leaf distances induced by that tree. The question is particularly topical since reliable estimates of genetic distance - even (if not in particular) by modern mass-sequencing methods - are, in general, available only for certain combinations of taxa.
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