Related papers: Deepest nodes in marked ordered trees
The degree distribution of an ordered tree $T$ with $n$ nodes is $\vec{n} = (n_0,\ldots,n_{n-1})$, where $n_i$ is the number of nodes in $T$ with $i$ children. Let $\mathcal{N}(\vec{n})$ be the number of trees with degree distribution…
We solve a recent question of Caro, Patk\'os and Tuza by determining the exact maximum number of edges in a bipartite connected graph as a function of the longest path it contains as a subgraph and of the number of vertices in each side of…
If trees are constructed from a pure birth process and one defines the depth of a leaf to be the number of edges to its root, it is known that the variance in the depth of a randomly selected leaf of a randomly selected tree grows linearly…
The properties of randomly evolving special trees having defined and analyzed already in two earlier papers (arXiv:cond-mat/0205650 and arXiv:cond-mat/0211092) have been investigated in the case when the continuous time parameter converges…
We focus on counting the number of labeled graphs on $n$ vertices and treewidth at most $k$ (or equivalently, the number of labeled partial $k$-trees), which we denote by $T_{n,k}$. So far, only the particular cases $T_{n,1}$ and $T_{n,2}$…
A treedepth decomposition of an undirected graph $G$ is a rooted forest $F$ on the vertex set of $G$ such that every edge $uv\in E(G)$ is in ancestor-descendant relationship in $F$. Given a weight function $w\colon V(G)\rightarrow…
For any positive integer $k$, there exist neural networks with $\Theta(k^3)$ layers, $\Theta(1)$ nodes per layer, and $\Theta(1)$ distinct parameters which can not be approximated by networks with $\mathcal{O}(k)$ layers unless they are…
The Rooted Maximum Leaf Outbranching problem consists in finding a spanning directed tree rooted at some prescribed vertex of a digraph with the maximum number of leaves. Its parameterized version asks if there exists such a tree with at…
We consider a Gibbs distribution over all spanning trees of an undirected, edge weighted finite graph, where, up to normalization, the probability of each tree is given by the product of its edge weights. Defining the weighted degree of a…
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…
Huynh, Joret, Micek, Seweryn, and Wollan (Combinatorica, 2022) introduced a graph parameter, later referred to as 2-treedepth and denoted $\mathrm{td}_2(\cdot)$. The parameter is the natural 2-connected version of treedepth. For every…
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…
The number of spanning trees in a graph $G$ is the total number of distinct spanning subgraphs of $G$ that are trees. In this paper we characterize the unique graph with a prescribed vertex (resp. edge) connectivity, minimum degree and…
Let $T$ be an $n$-node tree of maximum degree 4, and let $P$ be a set of $n$ points in the plane with no two points on the same horizontal or vertical line. It is an open question whether $T$ always has a planar drawing on $P$ such that…
Phylogenetic networks which are, as opposed to trees, suitable to describe processes like hybridization and horizontal gene transfer, play a substantial role in evolutionary research. However, while non-treelike events need to be taken into…
We propose a method for the classification of objects that are structured as random trees. Our aim is to model a distribution over the node label assignments in settings where the tree data structure is associated with node attributes…
Erd\H{o}s and Palka initiated the study of the maximal size of induced trees in random graphs in 1983. They proved that for every fixed $0<p<1$ the size of a largest induced tree in $G_{n,p}$ is concentrated around $2\log_q (np)$ with high…
We develop algorithms, implemented in Maple, that study the number of vertices with a particular number of children in a random ordered tree where all vertices must have a number of children in some finite set. By calculating the mixed…
We investigate the rank of the average mixing matrix of trees, with all eigenvalues distinct. The rank of the average mixing matrix of a tree on $n$ vertices with $n$ distinct eigenvalues is upper-bounded by $\frac{n}{2}$. Computations on…
A new 2-parameter family of central structures in trees, called central forests, is introduced. Minieka's $m$-center problem and McMorris's and Reid's central-$k$-tree can be seen as special cases of central forests in trees. A central…