Some families of trees arising in permutation analysis

TL;DR

This paper extends classical tree enumeration results to leaf-counted trees, applies them to strong interval trees with prime node restrictions, and analyzes permutation filtrations and their asymptotic behaviors.

Contribution

It introduces new enumeration techniques for leaf-counted trees and explores permutation filtrations via strong interval trees with prime node constraints.

Findings

Asymptotic enumeration of leaf-counted trees established

Permutation filtrations based on strong interval trees described

Analytic series convergence towards non-analytic limits demonstrated

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.