In mathematical phylogenetics, given a rooted binary leaf-labeled gene tree topology G and a rooted binary leaf-labeled species tree topology S with the same leaf labels, a coalescent history represents a possible mapping of the list of gene tree coalescences to the associated branches of the species tree on which those coalescences take place. For certain families of ordered pairs (G, S), the number of coalescent histories increases exponentially or even faster than exponentially with the number of leaves n. Other pairs have only a single coalescent history. We term a pair (G, S) lonely if it has only one coalescent history. Here, we characterize the set of all lonely pairs (G, S). Further, we characterize the set of pairs of rooted binary unlabeled tree shapes at least one of the labelings of which is lonely. We provide formulas for counting lonely pairs and pairs of unlabeled tree shapes with at least one lonely labeling. The lonely pairs provide a set of examples of pairs (G, S) for which the number of compact coalescent histories-which condense coalescent histories into a set of equivalence classes-is equal to the number of coalescent histories. Application of the condition that characterizes lonely pairs can also be used to reduce computation time for the enumeration of coalescent histories.

中文翻译：

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通过对足樱桃的方式枚举基因树和物种树的孤独对

在数学系统发育学中，给定有根二元叶标记基因树拓扑 G 和具有相同叶子标签的有根二元叶标记物种树拓扑 S，合并历史表示基因树合并列表到相关分支的可能映射发生这些合并的物种树。对于某些有序对 (G, S) 的家族，合并历史的数量随着叶子的数量 n 呈指数增长，甚至比指数增长更快。其他对只有一个合并历史。如果一对 (G, S) 只有一个合并历史，我们称它为孤独的。在这里，我们表征所有孤独对（G，S）的集合。此外，我们表征了一组有根的二元未标记树形状，其中至少有一个标记是孤独的。我们提供了计算孤独对和具有至少一个孤独标签的未标记树形状对的公式。孤独对提供了一组对 (G, S) 的示例，其中紧凑合并历史的数量（将合并历史压缩为一组等价类）等于合并历史的数量。应用表征孤独对的条件也可用于减少合并历史枚举的计算时间。