Related papers: On the Largest Common Subtree of Random Leaf-Label…

A binary tanglegram is a pair <S,T> of 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 or software engineering, it is required…

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 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…

There are several common ways to encode a tree as a matrix, such as the adjacency matrix, the Laplacian matrix (that is, the infinitesimal generator of the natural random walk), and the matrix of pairwise distances between leaves. Such…

Recently proposed budding tree is a decision tree algorithm in which every node is part internal node and part leaf. This allows representing every decision tree in a continuous parameter space, and therefore a budding tree can be jointly…

Minimum Bisection denotes the NP-hard problem to partition the vertex set of a graph into two sets of equal sizes while minimizing the width of the bisection, which is defined as the number of edges between these two sets. We first consider…

In this paper we study binary trees with choosable edge lengths, in particular rooted binary trees with the property that the two edges leading from every non-leaf to its two children are assigned integral lengths $l_1$ and $l_2$ with…

This paper investigates some properties of the number of subtrees of a tree with given degree sequence. These results are used to characterize trees with the given degree sequence that have the largest number of subtrees, which generalizes…

Connected acyclic graphs (trees) are data objects that hierarchically organize categories. Collections of trees arise in a diverse variety of fields, including evolutionary biology, public health, machine learning, social sciences and…

We investigate extremal statistical properties such as the maximal and the minimal heights of randomly generated binary trees. By analyzing the master evolution equations we show that the cumulative distribution of extremal heights…

A set ${\cal A} \subseteq \Set{1,...,N}$ is of type $B_2$ if all sums $a+b$, with $a\ge b$, $a,b\in {\cal A}$, are distinct. It is well known that the largest such set is of size asymptotic to $N^{1/2}$. For a $B_2$ set ${\cal A}$ of this…

We present an algorithm, called the Offset Tree, for learning to make decisions in situations where the payoff of only one choice is observed, rather than all choices. The algorithm reduces this setting to binary classification, allowing…

We prove that the treewidth of an Erd\"{o}s-R\'{e}nyi random graph $\rg{n, m}$ is, with high probability, greater than $\beta n$ for some constant $\beta > 0$ if the edge/vertex ratio $\frac{m}{n}$ is greater than 1.073. Our lower bound…

For a $k$-tree $T$, we prove that the maximum local mean order is attained in a $k$-clique of degree $1$ and that it is not more than twice the global mean order. We also bound the global mean order if $T$ has no $k$-cliques of degree $2$…

The problem of reconstructing evolutionary trees or phylogenies is of great interest in computational biology. A popular model for this problem assumes that we are given the set of leaves (current species) of an unknown binary tree and the…

A uniformly random graph on $n$ vertices with a fixed degree sequence, obeying a $\gamma$ subpower law, is studied. It is shown that, for $\gamma>3$, in a subcritical phase with high probability the largest component size does not exceed…

Let $\mathcal{T}_n$ be the set of all mappings $T:[n]\to[n]$, where $[n]=\{1,2,\ldots,n\}$. The corresponding graph $G_T$ of $T$, called a functional digraph, is a union of disjoint connected components. Each component is a directed cycle…

Let $A(n, d)$ denote the maximum size of a binary code of length $n$ and minimum Hamming distance $d$. Studying $A(n, d)$, including efforts to determine it as well to derive bounds on $A(n, d)$ for large $n$'s, is one of the most…

Knuth and Moore presented a theoretical lower bound on the number of leaves that any fixed-depth minimax tree-search algorithm traversing a uniform tree must explore, the so-called minimal tree. Since real-life minimax trees are not…

We investigate the randomized decision tree complexity of a specific class of read-once threshold functions. A read-once threshold formula can be defined by a rooted tree, every internal node of which is labeled by a threshold function…