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Attack trees are considered a useful tool for security modelling because they support qualitative as well as quantitative analysis. The quantitative approach is based on values associated to each node in the tree, expressing, for instance,…
Measuring the complexity of tree structures can be beneficial in areas that use tree data structures for storage, communication, and processing purposes. This complexity can then be used to compress tree data structures to their…
A tree is called k-ended tree if it has at most k leaves, where a leaf is a vertex of degree one. In this paper we prove that every 3-regular connected graph with n vertices such that n is greater than 8 has spanning sub tree with at most…
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…
Finite metric trees are known to have strict 1-negative type. In this paper we introduce a new family of inequalities that quantify the extent of the "strictness" of the 1-negative type inequalities for finite metric trees. These…
Regression trees and their ensemble methods are popular methods for nonparametric regression: they combine strong predictive performance with interpretable estimators. To improve their utility for locally smooth response surfaces, we study…
A quasiconformal tree is a doubling (compact) metric tree in which the diameter of each arc is comparable to the distance of its endpoints. We show that for each integer $n\geq 2$, the class of all quasiconformal trees with uniform branch…
A Conditional Tree Pattern (CTP) expands an XML tree pattern with labels attached to the descendant edges. These labels can be XML element names or Boolean CTPs. The meaning of a descendant edge labelled by A and ending in a node labelled…
Tree-structured neural networks encode a particular tree geometry for a sentence in the network design. However, these models have at best only slightly outperformed simpler sequence-based models. We hypothesize that neural sequence models…
We consider all spanning trees of a complete simple graph $\Gamma$ on $n$ vertices that contain a given $m-$forest $F$. We show that the number of such spanning trees, $\tau(F)$, doesn't depend on the structure of $F$ and is completely…
We prove that every oriented tree on $n$ vertices with bounded maximum degree appears as a spanning subdigraph of every directed graph on $n$ vertices with minimum semidegree at least $n/2+o(n)$. This can be seen as a directed graph…
Justification theory is an abstract unifying formalism that captures semantics of various non-monotonic logics. One intriguing problem that has received significant attention is the consistency problem: under which conditions are…
Tree Containment is a fundamental problem in phylogenetics useful for verifying a proposed phylogenetic network, representing the evolutionary history of certain species. Tree Containment asks whether the given phylogenetic tree (for…
In this paper, we investigate the structures of an extremal tree which has the minimal number of subtrees in the set of all trees with the given degree sequence of a tree. In particular, the extremal trees must be caterpillar and but in…
A graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers. We prove that for a given nullity more than 1, there are only finitely many integral trees. It is also shown that integral trees with…
Since they became observable, neuron morphologies have been informally compared with biological trees but they are studied by distinct communities, neuroscientists, and ecologists. The apparent structural similarity suggests there may be…
The aim of this note is to describe the structure of finite meadows. We will show that the class of finite meadows is the closure of the class of finite fields under finite products. As a corollary, we obtain a unique representation of…
A uniform recursive tree on $n$ vertices is a random tree where each possible $(n-1)!$ labeled recursive rooted tree is selected with equal probability. In this paper we introduce and study weighted trees, a non-uniform recursive tree model…
We consider the number of spanning trees in circulant graphs of $\beta n$ vertices with generators depending linearly on $n$. The matrix tree theorem gives a closed formula of $\beta n$ factors, while we derive a formula of $\beta-1$…
It has recently been shown that any simple (i.e. nonintersecting) polygonal chain in the plane can be reconfigured to lie on a straight line, and any simple polygon can be reconfigured to be convex. This result cannot be extended to tree…