On the number of increasing trees with label repetitions

Abstract

We study the asymptotic number of certain monotonically labeled increasing trees arising from a generalized evolution process. The main difference between the presented model and the classical model of binary increasing trees is that the same label can appear in distinct branches of the tree.In the course of the analysis we develop a method to extract asymptotic information on the coefficients of purely formal power series. The method is based on an approximate Borel transform (or, more generally, Mittag-Leffler transform) which enables us to quickly guess the exponential growth rate. With this guess the sequence is then rescaled and a singularity analysis of the generating function of the scaled counting sequence yields accurate asymptotics. The actual analysis is based on differential equations and a Tauberian argument.The counting problem for trees of size n exhibits interesting asymptotics involving powers of n with irrational exponents.

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

“…This work is a part of a long-term over-arching project, in which we aim at relaxing the classical rules of increasing labelling (described in, e.g., [BFS92]), by, for example, allowing labels to appear more than once in the tree. The following papers are part of this strand of research: [BGGW20,BGNS20] introduce and study models of label trees with less-constrained increasing labelling rules, but also other graphs structures like [BDF + 16] focuses on increasingly-labelled "diamonds" and [GGKW20] at a compacted structure that specifies classes of directed acyclic graphs.…”

Families of Monotonic Trees: Combinatorial Enumeration and Asymptotics

“…This example is of interest as it indicates where the change of asymptotic behavior occurs when one varies the allowed node arities, going from binary trees with label repetitions (see [4]) to weakly increasing Schröder trees (see [5]) (with arbitrary arity of nodes).…”

“…A related model, rooted increasing m-ary trees with label repetitions, also allowing multiple nodes to have the same label, appeared in [4] and can be seen as the foundations of our following new study that widely extends the latter model. These models of increasing trees are related to evolution processes in discrete time: starting from a single leaf, a tree is grown by selecting at each time-step a leaf and replacing it by an internal node to which new leaves are attached.…”

“…In our work we extend the study of [4], presenting a generic framework for an even more general class of combinatorial structures; in particular by relaxing the restrictions on node degrees (more precisely node out-degree in graph theory) and by allowing also for weakly increasing labeling sequences along branches of the trees. We study the following broad classes:…”

“…Firstly, we can interpret φ i , i ≥ 2, as the number of possible colors of a node of degree i. In this context the objects of study will correspond to the aforementioned strictly monotonic trees (these also include the so-called weakly increasing trees of [4,5]). Alternatively, the coefficients φ i , i ≥ 2, can be seen as the number of trees with i leaves belonging to some class of plane rooted unlabeled trees (in the sequel, we will refer to elements of such classes as tree-shapes).…”