In the context of reconstructing phylogenetic networks from a collection of phylogenetic trees, several characterisations and subsequently algorithms have been established to reconstruct a phylogenetic network that collectively embeds all trees in the input in some minimum way. For many instances however, the resulting network also embeds additional phylogenetic trees that are not part of the input. However, little is known about these inferred trees. In this paper, we explore the relationships among all phylogenetic trees that are embedded in a given phylogenetic network. First, we investigate some combinatorial properties of the collection P of all rooted binary phylogenetic trees that are embedded in a rooted binary phylogenetic network N. To this end, we associated a particular graph G, which we call rSPR graph, with the elements in P and show that, if |P|=2^k, where k is the number of vertices with in-degree two in N, then G has a Hamiltonian cycle. Second, by exploiting rSPR graphs and properties of hypercubes, we turn to the well-studied class of rooted binary level-1 networks and give necessary and sufficient conditions for when a set of rooted binary phylogenetic trees can be embedded in a level-1 network without inferring any additional trees. Lastly, we show how these conditions translate into a polynomial-time algorithm to reconstruct such a network if it exists.

翻译：在从系统发育树集合重建系统发育网络的背景下，已有若干刻画方法和相应算法被提出，旨在以某种最小方式重建一个能整体嵌入输入中所有树的系统发育网络。然而在许多实例中，所得网络还会额外嵌入不属于输入的系统发育树，但关于这些推断树的性质却知之甚少。本文探讨了给定系统发育网络中所嵌入的所有系统发育树之间的关系。首先，我们研究了嵌入在有根二叉树系统发育网络N中的所有有根二叉树系统发育树集合P的若干组合性质。为此，我们为P中的元素关联了一个特定图G（称为rSPR图），并证明：若|P|=2^k（其中k为网络N中入度为2的顶点数），则G存在哈密顿圈。其次，通过利用rSPR图与超立方体的性质，我们转向研究经典的有根系二叉树level-1网络类，给出了无需推断额外树即可将一组有根二叉树系统发育树嵌入level-1网络的充分必要条件。最后，我们展示了如何将这些条件转化为多项式时间算法（若该网络存在）以重建此类网络。