Related papers: A tropical interpretation of m-dissimilarity maps
In classification and forecasting with tabular data, one often utilizes tree-based models. Those can be competitive with deep neural networks on tabular data and, under some conditions, explainable. The explainability depends on the depth…
Given a tree of weighted vertices, it is sometimes possible to break the tree into two equally-weighted subtrees within an allowable error. We give a fast algorithm that finds an edge which breaks the tree into equal-weight components or…
We study the map degrees between quasitoric 4-manifolds. Our results rely on Theorems proved by Duan and Wang. We determine the set D (M, N) of all possible map degrees from M to N when M and N are certain quasitoric 4-manifolds. The…
We investigate an AND-OR tree T and a probability distribution d on the truth assignments to the leaves. Tarsi (1983) showed that if d is an independent and identical distribution (IID) such that probability of a leaf having value 0 is…
The subtrees and BC-subtrees (subtrees where any two leaves are at even distance apart) have been extensively studied in recent years. Such structures, under special constraints on degrees, have applications in many fields. Through an…
We present a simple and elementary procedure to sketch the tropical conic given by a degree--two homogeneous tropical polynomial. These conics are trees of a very particular kind. Given such a tree, we explain how to compute a defining…
We consider special cases of the two tree degree sequences problem. We show that if two tree degree sequences do not have common leaves then they always have edge-disjoint caterpillar realizations. By using a probabilistic method, we prove…
The interleaving distance is a key tool for comparing merge trees, which provide topological summaries of scalar functions. In this work, we define an average merge tree for a pair of merge trees using the interleaving distance. Since such…
Graphs are interesting structures: extremely useful to depict real-life problems, extremely easy to understand given a sketch, extremely complicated to represent formally, extremely complicated to compare. Phylogeny is the study of the…
Dual-tree algorithms are a widely used class of branch-and-bound algorithms. Unfortunately, developing dual-tree algorithms for use with different trees and problems is often complex and burdensome. We introduce a four-part logical split:…
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…
Trees or rooted trees have been generously studied in the literature. A forest is a set of trees or rooted trees. Here we give recurrence relations between the number of some kind of rooted forest with $k$ roots and that with $k+1$ roots on…
A "tree-partition" of a graph $G$ is a partition of $V(G)$ such that identifying the vertices in each part gives a tree. It is known that every graph with treewidth $k$ and maximum degree $\Delta$ has a tree-partition with parts of size…
A caterpillar tree is a connected, acyclic, graph in which all vertices are either a member of a central path, or joined to that central path by a single edge. In other words, caterpillar trees are the class of trees which become path…
Tree spanners approximate distances within graphs; a subtree of a graph is a tree $t$-spanner of the graph if and only if for every pair of vertices their distance in the subtree is at most $t$ times their distance in the graph. When a…
Let $M=(m_{ij})$ be a symmetric matrix of order $n$ whose elements lie in an arbitrary field $\mathbb{F}$, and let $G$ be the graph with vertex set $\{1,\ldots,n\}$ such that distinct vertices $i$ and $j$ are adjacent if and only if $m_{ij}…
A Yule tree is the result of a branching process with constant birth and death rates. Such a process serves as an instructive null model of many empirical systems, for instance, the evolution of species leading to a phylogenetic tree.…
Given a tropical divisor $D$ in the intersection of two tropical plane curves, we study when it can be realized as the tropicalization of the intersection of two algebraic curves, and give a sufficient condition. We show that under a…
Let $W_n$ denote the wheel graph having $n$-vertices. If $i$ and $j$ are any two vertices of $W_n$, define \[d_{ij}:= \begin{cases} 0 & \mbox{if}~i=j \\ 1 & \mbox{if}~i~ \mbox{and} ~j~ \mbox{are adjacent} \\ 2 & \mbox{else}. \end{cases}\]…
A tanglegram consists of two binary rooted trees with the same number of leaves and a perfect matching between the leaves of the trees. We show that the two halves of a random tanglegram essentially look like two independently chosen random…