Distributions of cherries and pitchforks for the Ford model

Limit theorems for patterns in ranked tree‐child networks

“…Next, the claimed expansion for Λ r ′ ,s ′ ,t ′ (n) follows by expanding 𝜏 n − 𝜏 n+1 , 𝜌 n − 𝜌 n+1 , and 𝜇 n − 𝜇 n+1 (see ( 5), (17), and (18), respectively; note that these expansions are all of the form ( 21)) and pointing out that the probabilities might contain factors of the form 𝜏 2 n , 𝜌 2 n , 𝜇 2 n , 𝜏 n 𝜌 n , 𝜏 n 𝜇 n , or 𝜌 n 𝜇 n which are however divided by n 2 and thus expanding them gives also terms of the form (21).…”

“…Studying properties of shape statistics for random models that are used to describe the evolutionary relationship between species is an important topic in biology. For phylogenetic trees , which are used to model nonreticulate evolution , many such studies have been performed and the stochastic behavior of, for example, pattern counts are known in great detail; see [4, 6–8, 15, 17, 19, 21, 22]. On the other hand, for phylogenetic networks , which are used to model reticulate evolution , very little is known about the number of occurrences of patterns when the networks from a given class are randomly sampled.…”