Families of Monotonic Trees: Combinatorial Enumeration and Asymptotics

Olivier Bodini (1) , Antoine Genitrini (2) , Mehdi Naima (1) , Alexandros Singh (1)

1 LIPN - Laboratoire d'Informatique de Paris-Nord
2 APR - Algorithmes, Programmes et Résolution

There exists a wealth of literature concerning families of increasing trees, particularly suitable for representing the evolution of either data structures in computer science, or probabilistic urns in mathematics, but are also adapted to model evolutionary trees in biology. The classical notion of increasing trees corresponds to labeled trees such that, along paths from the root to any leaf, node labels are strictly increasing; in addition nodes have distinct labels. In this paper we introduce new families of increasingly labeled trees relaxing the constraint of unicity of each label. Such models are especially useful to characterize processes evolving in discrete time whose nodes evolve simultaneously. In particular, we obtain growth processes for biology much more adequate than the previous increasing models. The families of monotonic trees we introduce are much more delicate to deal with, since they are not decomposable in the sense of Analytic Combinatorics. New tools are required to study the quantitative statistics of such families. In this paper, we first present a way to combinatorially specify such families through evolution processes, then, we study the tree enumerations.

Mots clés: Increasing trees; Monotonic trees; Borel transform; Evolution process; Analytic Combinatorics; Asymptotic enumeration

Domaines: Combinatoire [math.CO]; Algorithme et structure de données [cs.DS]

Olivier Bodini, Antoine Genitrini, Mehdi Naima, Alexandros Singh. Families of Monotonic Trees: Combinatorial Enumeration and Asymptotics. 15th International Computer Science Symposium, Jun 2020, Yekaterinburg, Russia. pp.155-168, ⟨10.1007/978-3-030-50026-9_11⟩. ⟨hal-03033670⟩