Related papers: Trees with power-like height dependent weight
We give two consistent constructions of trees $T$ whose finite power $T^{n+1}$ is sharply different from $T^n$: 1. An $\aleph_1$-tree $T$ whose interval topology $X_T$ is perfectly normal, but $(X_T)^2$ is not even countably metacompact. 2.…
The HOM problem, which asks whether the image of a regular tree language under a given tree homomorphism is again regular, is known to be decidable [Godoy & Gim\'enez: The HOM problem is decidable. JACM 60(4), 2013]. However, the problem…
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…
The spread of infectious disease in a human community or the proliferation of fake news on social media can be modeled as a randomly growing tree-shaped graph. The history of the random growth process is often unobserved but contains…
We consider the ensemble of N-dimensional random symmetric matrices A that have, in average, p non-zero elements per row. We study the asymptotic behavior of the norm of A in the limit of infinitely increasing N and p. We prove that the…
We study the probability density function $P(h_m,L)$ of the maximum relative height $h_m$ in a wide class of one-dimensional solid-on-solid models of finite size $L$. For all these lattice models, in the large $L$ limit, a central limit…
We consider sequences of finite weighted random graphs that converge locally to unimodular i.i.d. weighted random trees. When the weights are atomless, we prove that the matchings of maximal weight converge locally to a matching on the…
Panagiotou and Stufler recently proved an important fact on their way to establish the scaling limits of random P\'olya trees: a uniform random P\'olya tree of size $n$ consists of a conditioned critical Galton-Watson tree $C_n$ and many…
Binary search trees (BSTs) are fundamental data structures whose performance is largely governed by tree height. We introduce a block model for constructing BSTs by embedding internal BSTs into the nodes of an external BST -- a structure…
In this paper, we study the independence of shifts defined on $\mathbb{N}^d$ ($\mathbb{N}^d$ shift) and trees (tree-shift). Firstly, for the completeness of the article, we provide a proof that an $\mathbb{N}^d$ shift has positive…
$\beta(1,0)$-trees provide a convenient description of rooted non-separable planar maps. The involution $h$ on $\beta(1,0)$-trees was introduced to prove a complicated equidistribution result on a class of pattern-avoiding permutations. In…
We consider a random field $\phi(\mathbf{r})$ in $d$ dimensions which is largely concentrated around small `hotspots', with `weights', $w_i$. These weights may have a very broad distribution, such that their mean does not exist, or else is…
In this work, we investigate the spectrum of singularities of random stable trees with parameter $\gamma\in(1,2)$. We consider for that purpose the scaling exponents derived from two natural measures on stable trees: the local time $\ell^a$…
We investigate the effective resistance $R_n$ and conductance $C_n$ between the root and leaves of a binary tree of height $n$. In this electrical network, the resistance of each edge $e$ at distance $d$ from the root is defined by…
A tree-based network on a set $X$ of $n$ leaves is said to be universal if any rooted binary phylogenetic tree on $X$ can be its base tree. Francis and Steel showed that there is a universal tree-based network on $X$ in the case of $n=3$,…
A plane rooted tree is called a hipster tree if it has no nontrivial automorphisms. Equivalently, a tree is a hipster tree if no two siblings have isomorphic subtrees. We impose the hipster condition on various classes of rooted trees. By…
A fringe subtree of a rooted tree is a subtree consisting of one of the nodes and all its descendants. In this paper, we are specifically interested in the number of non-isomorphic trees that appear in the collection of all fringe subtrees…
We calculate the almost sure Hausdorff dimension for a general class of random affine planar code tree fractals. The set of probability measures describing the randomness includes natural measures in random $V$-variable and homogeneous…
We show that large critical multi-type Galton-Watson trees, when conditioned to be large, converge locally in distribution to an infinite tree which is analoguous to Kesten's infinite monotype Galton-Watson tree. This is proven when we…
We consider maps which are constructed from plane trees by assigning marks to the corners of each vertex and then connecting each pair of consecutive marks on their contour by a single edge. A measure is defined on the set of such maps by…