Combinatorial views on persistent characters in phylogenetics

Combinatorial perspectives on Dollo-$k$ characters in phylogenetics

“…Finally, note that the union of Dollo-0 and Dollo-1 characters corresponds to the set of so-called persistent characters, i.e. binary characters that can be realized on a rooted phylogenetic tree T by at most one 0 → 1 transition followed by at most one 1 → 0 transition (Bonizzoni et al, 2012;Wicke and Fischer, 2020). Our results for general Dollo-k characters will thus also shed new light on persistent characters.…”

“…More precisely, Dollo parsimony assumes that a trait may only be gained once but may be lost multiple times. It is less restrictive than the so-called perfect phylogeny model with persistent characters (Bonizzoni et al, 2012) that has recently attracted great attention in the literature (Bonizzoni et al, 2012) (see also Bonizzoni et al (2014Bonizzoni et al ( , 2016Bonizzoni et al ( , 2017a; Wicke and Fischer (2020)). In this model, a trait can both be gained and lost at most once.…”

“…In particular, we link and contrast general Dollo-k characters with the family of persistent characters (i.e. binary characters that can be realized on a rooted phylogenetic tree T by at most one gain and one loss along the edges of T ), for which a thorough combinatorial analysis was recently performed (Wicke and Fischer, 2020).…”

“…While this algorithm can also be used to decide whether a given character is persistent, we then show that Dollo-k characters and persistent characters are in general very different. First, while it was shown in (Wicke and Fischer, 2020) that there is a close relationship between the so-called Fitch algorithm (Fitch, 1971) and persistent characters, Dollo-k parsimony and Fitch parsimony are in general very different concepts. Second, while (Wicke and Fischer, 2020) also showed that there is a one-to-one correspondence between the balance of a tree (in terms of its Sackin index (Sackin, 1972)) and the number of persistent characters it induces, this correspondence does not hold for general Dollo-k characters.…”

“…First, while it was shown in (Wicke and Fischer, 2020) that there is a close relationship between the so-called Fitch algorithm (Fitch, 1971) and persistent characters, Dollo-k parsimony and Fitch parsimony are in general very different concepts. Second, while (Wicke and Fischer, 2020) also showed that there is a one-to-one correspondence between the balance of a tree (in terms of its Sackin index (Sackin, 1972)) and the number of persistent characters it induces, this correspondence does not hold for general Dollo-k characters. In fact, counting the number of Dollo-k characters for a given tree and fixed k is much more involved than counting the number of persistent characters, but we provide a recursive approach for it.…”