Related papers: $k$-protected vertices in binary search trees
We study rooted planar random trees with a probability distribution which is proportional to a product of weight factors $w_n$ associated to the vertices of the tree and depending only on their individual degrees $n$. We focus on the case…
Form a random k-SAT formula on n variables by selecting uniformly and independently m=rn clauses out of all 2^k (n choose k) possible k-clauses. The Satisfiability Threshold Conjecture asserts that for each k there exists a constant r_k…
We discuss the following question: given a finite BCK-algebra, what is the probability that two randomly selected elements commute? We call this probability the \textit{commuting degree} of a BCK-algebra. In a previous paper, the author…
Accessibility percolation is a new type of percolation problem inspired by evolutionary biology. To each vertex of a graph a random number is assigned and a path through the graph is called accessible if all numbers along the path are in…
We study the problem of computing the tightest upper and lower bounds on the probability that the sum of $n$ dependent Bernoulli random variables exceeds an integer $k$. Under knowledge of all pairs of bivariate distributions denoted by a…
Local convergence of bounded degree graphs was introduced by Benjamini and Schramm. This result was extended further by Lyons to bounded average degree graphs. In this paper we study the convergence of random tree sequences with given…
The "power of choice" has been shown to radically alter the behavior of a number of randomized algorithms. Here we explore the effects of choice on models of tree and network growth. In our models each new node has k randomly chosen…
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…
In a recent paper, the question of determining the fraction of binary trees that contain a fixed pattern known as the snowflake was posed. We show that this fraction goes to 1, providing two very different proofs: a purely combinatorial one…
We consider the decentralized binary hypothesis testing problem on trees of bounded degree and increasing depth. For a regular tree of depth t and branching factor k>=2, we assume that the leaves have access to independent and identically…
Let $T_n$ be a random recursive tree with $n$ nodes. List vertices of $T_n$ in decreasing order of degree as $v^1,\ldots,v^n$, and write $d^i$ and $h^i$ for the degree of $v^i$ and the distance of $v^i$ from the root, respectively. We prove…
The trie-based radix sort algorithm stores pairwise different infinite binary strings in the leaves of a binary tree in a way that the Ulam-Harris coding of each leaf equals a prefix (that is, an initial segment) of the corresponding…
We consider the minimum spanning tree problem on a weighted complete bipartite graph $K_{n_R, n_B}$ whose $n=n_R+n_B$ vertices are random, i.i.d. uniformly distributed points in the unit cube in $d$ dimensions and edge weights are the…
We study the fundamental question of how likely it is that two randomly chosen trees are isomorphic to each other for different models of random trees. We show that the probability decays exponentially for rooted labeled trees as well as…
In a one-parameter model for evolution of random trees, which also includes the Barabasi-Albert random tree, almost sure behavior and the limiting distribution of the degree of a vertex in a fixed position are examined. Results about Polya…
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…
An electrical network with the structure of a random tree is considered: starting from a root vertex, in one iteration each leaf (a vertex with zero or one adjacent edges) of the tree is extended by either a single edge with probability $p$…
We prove the existence of the local limit of uniform random d-regular bipartite planar maps, for every $d\geq 3$, as the number of vertices tends to infinity. The proof relies on a bijection between maps and so-called blossoming trees…
Thin spanning trees lie at the intersection of graph theory, approximation algorithms, and combinatorial optimization. They are central to the long-standing \emph{thin tree conjecture}, which asks whether every $k$-edge-connected graph…
Consider a rooted $N$-ary tree. To every vertex of this tree, we attach an i.i.d. continuous random variable. A vertex is called accessible if along its ancestral line, the attached random variables are increasing. We keep accessible…