Recently, the perfect phylogeny model with persistent characters has attracted great attention in the literature. It is based on the assumption that complex traits or characters can only be gained once and lost once in the course of evolution. Here, we consider a generalization of this model, namely Dollo parsimony, that allows for multiple character losses. More precisely, we take a combinatorial perspective on the notion of Dollo-k characters, i.e. traits that are gained at most once and lost precisely k times throughout evolution. We first introduce an algorithm based on the notion of spanning subtrees for finding a Dollo-k labelling for a given character and a given tree in linear time. We then compare persistent characters (consisting of the union of Dollo-0 and Dollo-1 characters) and general Dollo-k characters. While it is known that there is a strong connection between Fitch parsimony and persistent characters, we show that Dollo parsimony and Fitch parsimony are in general very different. Moreover, while it is known that there is a direct relationship between the number of persistent characters and the Sackin index of a tree, a popular index of tree balance, we show that this relationship does not generalize to Dollo-k characters. In fact, determining the number of Dollo-k characters for a given tree is much more involved than counting persistent characters, and we end this manuscript by introducing a recursive approach for the former. This approach leads to a polynomial time algorithm for counting the number of Dollo-k characters, and both this algorithm as well as the algorithm for computing Dollo-k labellings are publicly available in the Babel package for BEAST 2.

最近，具有持久特征的完美系统发育模型在文献中引起了极大的关注。它基于这样一个假设，即复杂的特征或性格在进化过程中只能获得一次而失去一次。在这里，我们考虑该模型的泛化，即 Dollo 简约，它允许多个字符丢失。更准确地说，我们对 Dollo- k字符的概念采取组合的观点，即在整个进化过程中最多获得一次而丢失恰好k次的特征。我们首先介绍一种基于生成子树概念的算法来寻找 Dollo- k在线性时间内标记给定字符和给定树。然后我们比较持久字符（由 Dollo-0 和 Dollo-1 字符的并集组成）和一般 Dollo- k字符。虽然已知 Fitch 简约与持久性字符之间存在很强的联系，但我们表明 Dollo 简约和 Fitch 简约大体上是非常不同的。此外，虽然已知持久字符的数量与树的 Sackin 指数（一种流行的树平衡指数）之间存在直接关系，但我们表明这种关系不能推广到 Dollo- k字符。实际上，确定 Dollo- k的数量给定树的字符比计算持久字符要复杂得多，我们通过介绍前者的递归方法来结束这篇手稿。这种方法导致了一种用于计算 Dollo- k字符数量的多项式时间算法，并且该算法以及计算 Dollo- k标签的算法都在 BEAST 2 的 Babel 包中公开可用。