The general problem of representing collections of trees as a single graph has led to many tree summary techniques. Many consensus approaches take sets of trees (either inferred as separate gene trees or gleaned from the posterior of a Bayesian analysis) and produce a single “best” tree. In scenarios where horizontal gene transfer or hybridization are suspected, networks may be preferred, which allow for nodes to have two parents, representing the fusion of lineages. One such construct is the cluster union network (CUN), which is constructed using the union of all clusters in the input trees. The CUN has a number of mathematically desirable properties, but can also present edges not observed in the input trees. In this paper we define a new network construction, the edge union network (EUN), which displays edges if and only if they are contained in the input trees. We also demonstrate that this object can be constructed with polynomial time complexity given arbitrary phylogenetic input trees, and so can be used in conjunction with network analysis techniques for further phylogenetic hypothesis testing.

中文翻译：

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使用边联合网络比较和显示系统发育树

将树的集合表示为单个图的一般问题导致了许多树摘要技术。许多共识方法采用树集（推断为单独的基因树或从贝叶斯分析的后验收集）并生成单个“最佳”树。在怀疑水平基因转移或杂交的情况下，网络可能是首选，它允许节点有两个父母，代表谱系的融合。一种这样的构造是集群联合网络 (CUN)，它是使用输入树中所有集群的联合构建的。CUN 具有许多数学上理想的属性，但也可以呈现输入树中未观察到的边缘。在本文中，我们定义了一个新的网络结构，边缘联合网络（EUN），当且仅当它们包含在输入树中时才显示边。我们还证明，在给定任意系统发育输入树的情况下，该对象可以用多项式时间复杂度构建，因此可以与网络分析技术结合使用，以进行进一步的系统发育假设检验。