Related papers: A note on high-dimensional discrepancy of subtrees

In the $L_0$ Fitting Tree Metrics problem, we are given all pairwise distances among the elements of a set $V$ and our output is a tree metric on $V$. The goal is to minimize the number of pairwise distance disagreements between the input…

Consider the nearest neighbor graph for the integer lattice Z^d in d dimensions. For a large finite piece of it, consider choosing a spanning tree for that piece uniformly among all possible subgraphs that are spanning trees. As the piece…

We study the problem of how well a tree metric is able to preserve the sum of pairwise distances of an arbitrary metric. This problem is closely related to low-stretch metric embeddings and is interesting by its own flavor from the line of…

The maximum common subtree isomorphism problem asks for the largest possible isomorphism between subtrees of two given input trees. This problem is a natural restriction of the maximum common subgraph problem, which is ${\sf NP}$-hard in…

Recently, deep neural networks have expanded the state-of-art in various scientific fields and provided solutions to long standing problems across multiple application domains. Nevertheless, they also suffer from weaknesses since their…

Orthology and paralogy relations are often inferred by methods based on gene similarity, which usually yield a graph depicting the relationships between gene pairs. Such relation graphs are known to frequently contain errors, as they cannot…

A metric phylogenetic tree relating a collection of taxa induces weighted rooted triples and weighted quartets for all subsets of three and four taxa, respectively. New intertaxon distances are defined that can be calculated from these…

Stanley asked whether a tree is determined up to isomorphism by its chromatic symmetric function. We approach Stanley's problem by studying the relationship between the chromatic symmetric function and other invariants. First, we prove…

Let $\T_{n}$ be the set of rooted labeled trees on $\set{0,...,n}$. A maximal decreasing subtree of a rooted labeled tree is defined by the maximal subtree from the root with all edges being decreasing. In this paper, we study a new…

The notion of tree entropy was introduced by the author as a normalized limit of the number of spanning trees in finite graphs, but is defined on random infinite rooted graphs. We give some new expressions for tree entropy; one uses…

For a graph $G=(V,E)$ and a set $S\subseteq V(G)$ of size at least $2$, an $S$-Steiner tree $T$ is a subgraph of $G$ that is a tree with $S\subseteq V(T)$. Two $S$-Steiner trees $T$ and $T'$ are internally disjoint (resp. edge-disjoint) if…

We show that for any fixed dense graph G and bounded-degree tree T on the same number of vertices, a modest random perturbation of G will typically contain a copy of T . This combines the viewpoints of the well-studied problems of embedding…

Let $T$ be a tree, we show that the null space of the adjacency matrix of $T$ has relevant information about the structure of $T$. We introduce the Null Decomposition of trees, and use it in order to get formulas for independence number and…

We show that the set of balanced binary trees is closed by interval in the Tamari lattice. We establish that the intervals [T, T'] where T and T' are balanced binary trees are isomorphic as posets to a hypercube. We introduce synchronous…

The mutational heterogeneity of tumours can be described with a tree representing the evolutionary history of the tumour. With noisy sequencing data there may be uncertainty in the inferred tree structure, while we may also wish to study…

We explicitly calculate the distance dependent correlation functions in a maximal entropy ensemble of random trees. We show that correlations remain disassortative at all distances and vanish only as a second inverse power of the distance.…

A transversal in a rooted tree is any set of nodes that meets every path from the root to a leaf. We let c(T,k) denote the number of transversals of size k in a rooted tree T. We define a partial order on the set of all rooted trees with n…

We compute the magnitude (an isometric invariant of metric spaces) of compact $\mathbb{R}$-trees and show that it equals $1 + L/2$, where $L \in [0, \infty]$ denotes the total length. Although length is the only geometric invariant captured…

In a rooted tree, we call a vertex {\em balanced} if it is at equal distance from all its descendant leaves. We count balanced vertices in three different tree varieties. For decreasing binary trees, we can prove that the probability that a…

The tree-metric theorem provides a necessary and sufficient condition for a dissimilarity matrix to be a tree metric, and has served as the foundation for numerous distance-based reconstruction methods in phylogenetics. Our main result is…