The so-called binary perfect phylogeny with persistent characters has recently been thoroughly studied in computational biology as it is less restrictive than the well known binary perfect phylogeny. Here, we focus on the notion of (binary) persistent characters, i.e. characters that can be realized on a phylogenetic tree by at most one $0 \rightarrow 1$ transition followed by at most one $1 \rightarrow 0$ transition in the tree, and analyze these characters under different aspects. First, we illustrate the connection between persistent characters and Maximum Parsimony, where we characterize persistent characters in terms of the first phase of the famous Fitch algorithm. Afterwards we focus on the number of persistent characters for a given phylogenetic tree. We show that this number solely depends on the balance of the tree. To be precise, we develop a formula for counting the number of persistent characters for a given phylogenetic tree based on an index of tree balance, namely the Sackin index. Lastly, we consider the question of how many (carefully chosen) binary characters together with their persistence status are needed to uniquely determine a phylogenetic tree and provide an upper bound for the number of characters needed.

中文翻译：

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系统发育学中持久性特征的组合观点

具有持久特征的所谓二元完美系统发育最近在计算生物学中得到了彻底研究，因为它比众所周知的二元完美系统发育限制要少。在这里，我们关注（二进制）持久字符的概念，即可以在系统发育树上通过最多一个 $0 \rightarrow 1$ 转换然后在树中最多一个 $1 \rightarrow 0$ 转换来实现的字符，以及从不同的方面分析这些特征。首先，我们说明了持久字符和最大简约之间的联系，我们根据著名的 Fitch 算法的第一阶段来表征持久字符。之后，我们关注给定系统发育树的持久特征的数量。我们表明这个数字完全取决于树的平衡。准确地说，我们开发了一个公式，用于根据树平衡指数（即 Sackin 指数）计算给定系统发育树的持久性特征数。最后，我们考虑需要多少（精心选择的）二进制字符及其持久状态来唯一确定系统发育树并提供所需字符数的上限的问题。