Abstract
We extend classical results on simple varieties of trees (asymptotic enumeration, average behavior of tree parameters) to trees counted by their number of leaves. Motivated by genome comparison of related species, we then apply these results to strong interval trees with a restriction on the arity of prime nodes. Doing so, we describe a filtration of the set of permutations based on their strong interval trees. This filtration is also studied from a purely analytical point of view, thus illustrating the convergence of analytic series towards a non-analytic limit at the level of the asymptotic behavior of their coefficients.
Highlights
Permutations can be realized in many different forms using a variety of structures
Bouvel et al [7] considered a subclass of strong interval trees – selected because they represent what is known as commuting scenarios [3] – that correspond to the class of separable permutations
Bouvel et al [7] conclude their study on separable permutations with a suggestion for the class to study: strong interval trees with degree restrictions on certain internal nodes. These trees offer a very controlled way to introduce bias in the distribution of strong interval trees. This is precisely what we do in this work: we study strong interval trees where the so-called prime nodes have a bounded number of children
Summary
Permutations can be realized in many different forms using a variety of structures. The idea of viewing permutations as enriched trees has been around for several decades in different research communities. By studying asymptotic enumeration and parameter formulas for separable permutations, they proved that the complexity of the algorithm of [3] solving the perfect sorting by reversals problem is polynomial-time on separable permutations, whereas this problem is NP-complete in general We give a complete analysis of these restricted sets of trees They can be completely understood combinatorially and analytically, and so we have access to formulas for enumeration and the average values of some tree parameters that are related to computing perfect sorting scenarios. This restriction is motivated by algorithmic problems in genome rearrangements. A preliminary version of this work appeared in the extended abstract [8]
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