Related papers: Tree evolution processes for bucket increasing tre…
The goal of these lectures is to review some mathematical aspects of random tree models used in evolutionary biology to model gene trees or species trees. We start with stochastic models of tree shapes (finite trees without edge lengths),…
We extend classical results on simple varieties of trees (asymptotic enumeration, average behavior of tree parameters) to trees counted by their number of leaves. Motivated by genome comparison of related species, we then apply these…
Phylogenetic trees are binary nonplanar trees with labelled leaves, and plane oriented recursive trees are planar trees with an increasing labelling. Both families are enumerated by double factorials. A bijection is constructed, using the…
We consider linear preferential attachment trees, and show that they can be regarded as random split trees in the sense of Devroye (1999), although with infinite potential branching. In particular, this applies to the random recursive tree…
The search for similarity and dissimilarity measures on phylogenetic trees has been motivated by the computation of consensus trees, the search by similarity in phylogenetic databases, and the assessment of clustering results in…
As a unification of increasing trees and plane trees, the weakly increasing trees labeled by a multiset was introduced by Lin-Ma-Ma-Zhou in 2021. Motived by some symmetries in plane trees proved recently by Dong, Du, Ji and Zhang, we…
The main purpose of this paper is to consider the multiple birth properties for multi-type Markov branching processes. We first construct a new multi-dimensional Markov process based on the multi-type Markov branching process, which can…
Additive models, such as produced by gradient boosting, and full interaction models, such as classification and regression trees (CART), are widely used algorithms that have been investigated largely in isolation. We show that these models…
Motivated by the problem of maintaining data structures for a large sets of points that are evolving over the course of time, we consider the problem of maintaining a set of labels assigned to the vertices of a tree, where the locations of…
This paper considers the enumeration of ternary trees (i.e. rooted ordered trees in which each vertex has 0 or 3 children) avoiding a contiguous ternary tree pattern. We begin by finding recurrence relations for several simple tree…
Certain classes of problems, including perceptual data understanding, robotics, discovery, and learning, can be represented as incremental, dynamically constructed belief networks. These automatically constructed networks can be dynamically…
We evolve binary mux-6 trees for up to 100000 generations evolving some programs with more than a hundred million nodes. Our unbounded Long-Term Evolution Experiment LTEE GP appears not to evolve building blocks but does suggests a limit to…
In order to speed-up classification models when facing a large number of categories, one usual approach consists in organizing the categories in a particular structure, this structure being then used as a way to speed-up the prediction…
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…
We consider the lower-triangular matrix of generating polynomials that enumerate $k$-component forests of rooted trees on the vertex set $[n]$ according to the number of improper edges (generalizations of the Ramanujan polynomials). We show…
Generating mathematical equations from natural language requires an accurate understanding of the relations among math expressions. Existing approaches can be broadly categorized into token-level and expression-level generation. The former…
For a labelled tree on the vertex set $[n]:=\{1,2,..., n\}$, define the direction of each edge $ij$ to be $i\to j$ if $i<j$. The indegree sequence of $T$ can be considered as a partition $\lambda \vdash n-1$. The enumeration of trees with a…
Working with generating functions, the combinatorics of a recurrence relation can be expressed in a way that allows for more efficient calculation of the quantity. This is true of the Catalan numbers for an ordered binary tree…
We consider packing tree degree sequences in this paper. We set up a conjecture that any arbitrary number of tree degree sequences without common leaves have edge disjoint tree realizations. This conjecture is known to be true for $2$ and…
Current-day genomes bear the mark of the evolutionary processes. One of the strongest indications is the sequence homology among families of proteins that perform similar biological functions in different species. The number of proteins in…