Ranked Schröder Trees

In biology, a phylogenetic tree is a tool to represent the evolutionary relationship between species. Unfortunately, the classical Schröder tree model is not adapted to take into account the chronology between the branching nodes. In particular, it does not answer the question: how many different phylogenetic stories lead to the creation of n species and what is the average time to get there? In this paper, we enrich this model in two distinct ways in order to obtain two ranked tree models for phylogenetics, i.e. models coding chronology. For that purpose, we first develop a model of (strongly) increasing Schröder trees, symbolically described in the classical context of increasing labeling. Then we introduce a generalization for the labeling with some unusual order constraint in Analytic Combinatorics (namely the weakly increasing trees). Although these models are direct extensions of the Schröder tree model, it appears that they are also in one-to-one correspondence with several classical combinatorial objects. Through the paper, we present these links, exhibit some parameters in typical large trees and conclude the studies with efficient uniform samplers.