Related papers: The height of random binary unlabelled trees
We give a counterexample to the conjecture of Martin and Thatte that two balanced rooted binary leaf-labelled trees on $n$ leaves have a maximum agreement subtree (MAST) of size at least $n^{\frac{1}{2}}$. In particular, we show that for…
In this paper we consider the enumeration of binary trees avoiding non-contiguous binary tree patterns. We begin by computing closed formulas for the number of trees avoiding a single binary tree pattern with 4 or fewer leaves and compare…
In the longest plane spanning tree problem, we are given a finite planar point set $\mathcal{P}$, and our task is to find a plane (i.e., noncrossing) spanning tree for $\mathcal{P}$ with maximum total Euclidean edge length. Despite more…
We study the following growth model on a regular d-ary tree. Points at distance n adjacent to the existing subtree are added with probabilities proportional to alpha^{-n}, where alpha<1 is a positive real parameter. The heights of these…
We prove empirical central limit theorems for the distribution of levels of various random fields defined on high-dimensional discrete structures as the dimension of the structure goes to $\infty$. The random fields considered include costs…
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…
Geometry of networks endowed with a causal structure is discussed using the conventional framework of equilibrium statistical mechanics. The popular growing network models appear as particular causal models. We focus on a class of tree…
Let ${\mathbf T}_n$ be a uniformly random tree with vertex set $[n]=\{1,\ldots,n\}$, let $\Delta_{{\mathbf T}_n}$ be the largest vertex degree in ${\mathbf T}_n$, and let $\lambda_1({\mathbf T}_n),\ldots,\lambda_n({\mathbf T}_n)$ be the…
We investigate the number of permutations that occur in random labellings of trees. This is a generalisation of the number of subpermutations occurring in a random permutation. It also generalises some recent results on the number of…
We study the height of the delta peak at 0 in the spectrum of random tree incidence matrices. We show that the average fraction of the spectrum occupied by the eigenvalue 0 in a large random tree is asymptotic to 2x-1 =…
The early development of a zygote can be mathematically described by a developmental tree. To compare developmental trees of different species, we need to define distances on trees. If children cells after a division are not…
The recursive and hierarchical structure of full rooted trees is applicable to represent statistical models in various areas, such as data compression, image processing, and machine learning. In most of these cases, the full rooted tree is…
A weighted recursive tree is an evolving tree in which vertices are assigned random vertex-weights and new vertices connect to a predecessor with a probability proportional to its weight. Here, we study the maximum degree and near-maximum…
When using machine learning for imbalanced binary classification problems, it is common to subsample the majority class to create a (more) balanced training dataset. This biases the model's predictions because the model learns from data…
We study the scaling limit of random forest with prescribed degree sequence in the regime that the largest tree consists of all but a vanishing fraction of nodes. We give a description of the limit of the forest consisting of the small…
Exploratory data analysis is crucial for developing and understanding classification models from high-dimensional datasets. We explore the utility of a new unsupervised tree ensemble called uncharted forest for visualizing class…
Given a Galton-Watson process conditioned to have total progeny equal to $n$, we study the asymptotic probability that this conditioned Galton-Watson process has distance to the border bigger or equal than $k$, as the number of nodes $n…
We give a unified treatment of the limit, as the size tends to infinity, of simply generated random trees, including both the well-known result in the standard case of critical Galton--Watson trees and similar but less well-known results in…
We prove local convergence results for the uniformly random, labelled or unlabelled, graphs from subcritical families. As an example special case, we prove Benjamini-Schramm convergence for the uniform random unlabelled tree. We introduce a…
Let $T$ be a random tree taken uniformly at random from the family of labelled trees on $n$ vertices. In this note, we provide bounds for $c(n)$, the number of sub-trees of $T$ that hold asymptotically almost surely. With computer support…