Strict monotonic trees arising from evolutionary processes: Combinatorial and probabilistic study

In this paper we study two models of labelled random trees that generalise the original unlabelled Schröder tree. Our new can be seen as for phylogenetic in which nodes represent species and labels encode order appearance these species, thus chronology evolution. One important feature our is they generated efficiently thanks to a dynamical, recursive construction. first model an increasing tree classical sense (labels increase along each branch label appears only once). To better trees, relax rules labelling by allowing repetitions second model. For models, provide asymptotic theorems different characteristics (e.g. degree root, distribution, height, etc.), giving extensive information about typical shapes trees. We also efficient algorithms generate large models. The proofs are based on combination analytic combinatorics, probabilistic methods, bijective methods (we exhibit bijections between well-known literature such permutations Stirling numbers both kinds). It turns out even though labelled, specified simply world ordinary generating functions. However, resulting functions will formal. Then, applying Borel transforms amenable techniques combinatorics.