Related papers: 0-1 laws for pattern occurrences in phylogenetic t…
Let $L(n,d)$ denote the minimum possible number of leaves in a tree of order $n$ and diameter $d.$ In 1975 Lesniak gave the lower bound $B(n,d)=\lceil 2(n-1)/d\rceil$ for $L(n,d).$ When $d$ is even, $B(n,d)=L(n,d).$ But when $d$ is odd,…
The Ward numbers $W(n,k)$ combinatorially enumerate set partitions with block sizes $\geq 2$ and phylogenetic trees (total partition trees). We prove that $W(n,k)$ also counts \emph{increasing Schr\"oder trees} by verifying they satisfy…
Following Poupard's study of strictly ordered binary trees with respect to two parameters, namely, "end of minimal chain" and "parent of maximum leaf" a true Tree Calculus is being developed to solve a partial difference equation system and…
In evolutionary biology, networks are becoming increasingly used to represent evolutionary histories for species that have undergone non-treelike or reticulate evolution. Such networks are essentially directed acyclic graphs with a leaf set…
Decision trees and diffusion models are ostensibly disparate model classes, one discrete and hierarchical, the other continuous and dynamic. This work unifies the two by establishing a crisp mathematical correspondence between hierarchical…
We introduce a random graph model based on k-trees, which can be generated by applying a probabilistic preferential attachment rule, but which also has a simple combinatorial description. We carry out a precise distributional analysis of…
New bounds on the number of similar or directly similar copies of a pattern within a finite subset of the line or the plane are proved. The number of equilateral triangles whose vertices all lie within an $n$-point subset of the plane is…
We present a new characterization of $k$-trees based on their reduced clique graphs and $(k+1)$-line graphs, which are block graphs. We explore structural properties of these two classes, showing that the number of clique-trees of a…
A rooted acyclic digraph N with labelled leaves displays a tree T when there exists a way to select a unique parent of each hybrid vertex resulting in the tree T. Let Tr(N) denote the set of all trees displayed by the network N. In general,…
Phylogenetic (i.e. leaf-labeled) trees play a fundamental role in evolutionary research. A typical problem is to reconstruct such trees from data like DNA alignments (whose columns are often referred to as characters), and a simple…
We study the leaf-to-leaf distances on full and complete m-ary graphs using a recursive approach. In our formulation, leaves are ordered along a line. We find explicit analytical formulae for the sum of all paths for arbitrary leaf-to-leaf…
Limiting distributions are derived for the sparse connected components that are present when a random graph on $n$ vertices has approximately $\half n$ edges. In particular, we show that such a graph consists entirely of trees, unicyclic…
Neutral macroevolutionary models, such as the Yule model, give rise to a probability distribution on the set of discrete rooted binary trees over a given leaf set. Such models can provide a signal as to the approximate location of the root…
A split tree of cardinality $n$ is constructed by distributing $n$ "balls" in a subset of vertices of an infinite tree which encompasses many types of random trees such as $m$-ary search trees, quad trees, median-of-$(2k+1)$ trees,…
The subject of pattern avoiding permutations has its roots in computer science, namely in the problem of sorting a permutation through a stack. A formula for the number of permutations of length n that can be sorted by passing it twice…
In 1986, Janson showed that the number of edges in the union of $k$ random spanning trees in the complete graph $K_n$ is a shifted Poisson distribution. Using results from the theory of electrical networks, we provide a new proof of this…
A tree-based network on a set $X$ of $n$ leaves is said to be universal if any rooted binary phylogenetic tree on $X$ can be its base tree. Francis and Steel showed that there is a universal tree-based network on $X$ in the case of $n=3$,…
Driven by the need for better models that allow one to shed light into the question how life's diversity has evolved, phylogenetic networks have now joined phylogenetic trees in the center of phylogenetics research. Like phylogenetic trees,…
In their study of the giant component in inhomogeneous random graphs, Bollob\'as, Janson, and Riordan introduced a class of branching processes parametrized by a possibly unbounded graphon. We prove that the tree structures underlying two…
We derive asymptotics of moments and identify limiting distributions, under the random permutation model on m-ary search trees, for functionals that satisfy recurrence relations of a simple additive form. Many important functionals…