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We present families of combinatorial classes described as trees with nodes that can carry one of two types of "flowers": integer partitions or integer compositions. Two parameters on the flowers of trees will be considered: the number of…
This work focuses on clustering populations with a hierarchical dependency structure that can be described by a tree. A particular example that is the focus of our work is the phylogenetic tree, with nodes often representing biological…
Staged tree models are a discrete generalization of Bayesian networks. We show that these form curved exponential families and derive their natural parameters, sufficient statistic, and cumulant-generating function as functions of their…
The use of machine learning algorithms in finance, medicine, and criminal justice can deeply impact human lives. As a consequence, research into interpretable machine learning has rapidly grown in an attempt to better control and fix…
We consider exact enumerations and probabilistic properties of ranked trees when generated under the random coalescent process. Using a new approach, based on generating functions, we derive several statistics such as the exact probability…
This article presents a unified bijective scheme between planar maps and blossoming trees, where a blossoming tree is defined as a spanning tree of the map decorated with some dangling half-edges that enable to reconstruct its faces. Our…
Several real-world and abstract structures and systems are characterized by marked hierarchy to the point of being expressed as trees. Because the study of these entities often involves sampling (or discovering) the tree nodes in a specific…
A growing random graph is constructed by successively sampling without replacement an element from the pool of virtual vertices and edges. At start of the process the pool contains $N$ virtual vertices and no edges. Each time a vertex is…
We present a bijective algorithm with which an arbitrary permutation decomposes canonically into elementary blocks which we call families, which are sets with a specified number of ascents and descents. We show that families, arranged in an…
We consider the process of uncovering the vertices of a random labeled tree according to their labels. First, a labeled tree with $n$ vertices is generated uniformly at random. Thereafter, the vertices are uncovered one by one, in order of…
We present here a new and universal approach for the study of random and/or trees, unifying in one framework many different models, including some novel ones not yet understood in the literature. An and/or tree is a Boolean expression…
We study various classes of random processes defined on the regular tree $T_d$ that are invariant under the automorphism group of $T_d$. Most important ones are factor of i.i.d. processes (randomized local algorithms), branching Markov…
The in-order traversal provides a natural correspondence between binary trees with a decreasing vertex labeling and endofunctions on a finite set. By suitably restricting the vertex labeling we arrive at a class of trees that we call…
Theoretical background and an implementation of the (p)-group generation algorithm by Newman and O'Brien are used to provide computational evidence of a new type of periodically repeating patterns in pruned descendant trees of finite…
The growth of ballistic aggregates on deterministic fractal substrates is studied by means of numerical simulations. First, we attempt the description of the evolving interface of the aggregates by applying the well-established…
The determination of cluster centers generally depends on the scale that we use to analyze the data to be clustered. Inappropriate scale usually leads to unreasonable cluster centers and thus unreasonable results. In this study, we first…
We introduce bijections between families of rooted maps with unfixed genus and families of so-called blossoming trees endowed with an arbitrary forward matching of their leaves. We first focus on Eulerian maps with controlled vertex…
We introduce a non-increasing tree growth process $((T_n,\sigma_n),\, n\ge 1)$, where $T_n$ is a rooted labeled tree on $n$ vertices and ${\sigma}_n$ is a permutation of the vertex labels. The construction of $(T_{n},{\sigma}_n)$ from…
Internal diffusion-limited aggregation is a growth model based on random walk in Z^d. We study how the shape of the aggregate depends on the law of the underlying walk, focusing on a family of walks in Z^2 for which the limiting shape is a…
We construct random locally compact real trees called Levy trees that are the genealogical trees associated with continuous-state branching processes. More precisely, we define a growing family of discrete Galton-Watson trees with i.i.d.…