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

Abstract

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A Combinatorial Link Between Labelled Graphs and Increasingly Labelled Schröder Trees

“…Proof. The proof for this theorem is analogous to the one for Theorem 3.6.5 in [3] after showing that both UnrankBinomial and UnrankComposition run in Θ(n) in the number of arithmetic operations. Using F , substitute in T , using any traversal, the leaves selected with B with 27: internal nodes and new leaves according to C; the other leaves are changed as 28:…”

“…Unranking algorithmic has been developed in the 70's by Nijenhuis and Wilf [21] and then has been introduced to the context of analytic combinatorics by Martínez and Molinero [18]. We use the same method as the one described in [3].…”

“…Acknowledgement. The authors thank Cécile Mailler for fruitful discussions to relate this model of Schröder trees to the ones presented in [3]. The authors thank the anonymous referees for their comments and suggested improvements.…”

“…In the recent years, many links were found between evolution processes in the form of increasing trees and classical combinatorial structures, for instance permutations are known to be in bijection with increasing binary trees [10, p. 143], increasing even trees and alternating permutations are put in bijection in [16,5], plane recursive trees are related to Stirling permutations [15] and more recently increasing Schröder trees have been proved in one-to-one correspondence with even permutations and with weak orderings on sets of n elements (counted by ordered Bell numbers) in [2,3]. By adding some constraint in the increasing labelling of the latter model, Lin et al [17] exhibited closed relationships between various families of polynomials (especially Eulerian, Narayana and Savage and Schuster polynomials).…”

“…The trees he studied were counted by their number of leaves which represent the number of species. We pursue enriching Schröder trees in the same vein as [2,3] but with a more general model.…”