Related papers: On symmetries in phylogenetic trees
In biology, a phylogenetic tree is a tool to represent the evolutionary relationship between species. Unfortunately, the classical Schr\"oder tree model is not adapted to take into account the chronology between the branching nodes. In…
We study the distribution of fringe trees in Patricia tries (extending earlier results by Ischebeck (2025)) and compressed binary search trees; both cases are random binary trees that have been compressed by deleting nodes of outdegree 1 so…
A compacted binary tree is a graph created from a binary tree such that repeatedly occurring subtrees in the original tree are represented by pointers to existing ones, and hence every subtree is unique. Such representations form a special…
A genus one labeled circle tree is a tree with its vertices on a circle, such that together they can be embedded in a surface of genus one, but not of genus zero. We define an e-reduction process whereby a special type of subtree, called an…
A group is boundedly simple if, for some constant N, every nontrivial conjugacy class generates the whole group in N steps. For a large class of trees, Tits proved simplicity of a canonical subgroup of the automorphism group, which is…
We consider the classical Smoluchowski coagulation equation with a general frequency kernel. We show that there exists a natural deterministic solution expansion in the non-associative algebra generated by the convolution product of the…
Trees are useful entities allowing to model data structures and hierarchical relationships in networked decision systems ubiquitously. An ordered tree is a rooted tree where the order of the subtrees (children) of a node is significant. In…
A classical result, fundamental to evolutionary biology, states that an edge-weighted tree $T$ with leaf set $X$, positive edge weights, and no vertices of degree 2 can be uniquely reconstructed from the set of leaf-to-leaf distances…
A tree-based network $\mathcal N$ on $X$ is universal if every rooted binary phylogenetic $X$-tree is a base tree for $\mathcal N$. Hayamizu and, independently, Zhang constructively showed that, for all positive integers $n$, there exists…
We show that for any two values $\alpha, \beta >0 $ for which $\alpha+\beta>1$ then there is a value $N$ so that for all $n \geq N$ the following holds. For any binary phylogenetic tree $T$ on $n$ leaves there is a set of $\lfloor n^\alpha…
A \emph{binary tanglegram} is a drawing of a pair of rooted binary trees whose leaf sets are in one-to-one correspondence; matching leaves are connected by inter-tree edges. For applications, for example, in phylogenetics, it is essential…
Let $\mathfrak{g}$ be a semisimple complex Lie algebra of finite dimension and $\mathfrak{h}$ be a semisimple subalgebra. We present an approach to find the branching rules for the pair $\mathfrak{g}\supset\mathfrak{h}$. According to an…
We consider the generating polynomial of the number of rooted trees on the set $\{1,2,\dots,n\}$ counted by the number of descending edges (a parent with a greater label than a child). This polynomial is an extension of the descent…
In the symmetric group on a set of size 2n, let P_{2n} denote the conjugacy class of involutions with no fixed points (equivalently, we refer to these as ``pairings'', since each disjoint cycle has length 2). Harer and Zagier explicitly…
Consider a random real tree whose leaf set, or boundary, is endowed with a finite mass measure. Each element of the tree is further given a type, or allele, inherited from the most recent atom of a random point measure…
Tree alignment graphs (TAGs) provide an intuitive data structure for storing phylogenetic trees that exhibits the relationships of the individual input trees and can potentially account for nested taxonomic relationships. This paper…
The purpose of this paper is to analyze certain statistics of a recently introduced non-uniform random tree model, biased recursive trees. This model is based on constructing a random tree by establishing a correspondence with non-uniform…
One strategy for reconstruction of phylogenetic networks is to solve the phylogenetic network problem, which involves inferring phylogenetic trees first and subsequently computing the smallest phylogenetic network that displays all the…
The algebraic properties of flattenings and subflattenings provide direct methods for identifying edges in the true phylogeny -- and by extension the complete tree -- using pattern counts from a sequence alignment. The relatively small…
We study a random fragmentation process and its associated random tree. The process has earlier been studied by Dean and Majumdar (J. Phys. A: Math. Gen., vol. 35, L501--L507), who found a phase transition: the number of fragmentations is…