Related papers: Injective Labeled Oriented Trees are Aspherical
We define decorated $\alpha$-stable trees which are informally obtained from an $\alpha$-stable tree by blowing up its branchpoints into random metric spaces. This generalizes the $\alpha$-stable looptrees of Curien and Kortchemski, where…
We give a representation-theoretic bijection between rooted labeled forests with $n$ vertices and complete exceptional sequences for the quiver of type $A_n$ with straight orientation. The ascending and descending vertices in the forest…
We provide a short combinatorial proof of Cayley's formula by means of a bijective map to an outcome space of an urn-drawing problem. Furthermore we introduce an algebraic structure on the set of labeled trees, which provides a more…
A number of hook formulas and hook summation formulas have previously appeared, involving various classes of trees. One of these classes of trees is rooted trees with labelled vertices, in which the labels increase along every chain from…
We give a representation for labeled ordered trees that supports labeled queries such as finding the i-th ancestor of a node with a given label. Our representation is succinct, namely the redundancy is small-o of the optimal space for…
A graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers. Recently, Csikvari proved the existence of integral trees of any even diameter. In the odd case, integral trees have been constructed with…
A uniform attachment tree is a random tree that is generated dynamically. Starting from a fixed "seed" tree, vertices are added sequentially by attaching each vertex to an existing vertex chosen uniformly at random. Upon observing a large…
A staged tree model is a discrete statistical model encoding relationships between events. These models are realised by directed trees with coloured vertices. In algebro-geometric terms, the model consists of points inside a toric variety.…
A tree $T$ is said to be homogeneous if it is uniquely rooted and there exists an integer $b\meg 2$, called the branching number of $T$, such that every $t\in T$ has exactly $b$ immediate successors. A vector homogeneous tree $\mathbf{T}$…
The history of gene families -- which are equivalent to event-labeled gene trees -- can to some extent be reconstructed from empirically estimated evolutionary event-relations containing pairs of orthologous, paralogous or xenologous genes.…
We present a new probabilistic proof of Otter's asymptotic formula for the number of unlabelled trees with a given number of vertices. We additionally prove a new approximation result, showing that the total variation distance between…
Merge trees are a type of graph-based topological summary that tracks the evolution of connected components in the sublevel sets of scalar functions. They enjoy widespread applications in data analysis and scientific visualization. In this…
Graceful tree conjecture is a well-known open problem in graph theory. Here we present a computational approach to this conjecture. An algorithm for finding graceful labelling for trees is proposed. With this algorithm, we show that every…
We give an explicit combinatorial proof of a weighted version of strong log-concavity for the generating polynomial of increasing spanning forests of a finite simple graph equipped with a total ordering of the vertices. In contrast to…
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…
Using Machine Learning systems in the real world can often be problematic, with inexplicable black-box models, the assumed certainty of imperfect measurements, or providing a single classification instead of a probability distribution. This…
Cayley's formula states that the number of labelled trees on $n$ vertices is $n^{n-2}$, and many of the current proofs involve complex structures or rigorous computation. We present a bijective proof of the formula by providing an…
A graph $G$ on $m$ edges is graceful if there is an injection $f : V(G) \to \{0, 1, \ldots, m\}$ whose induced edge labels $\{|f(u) - f(v)| : uv \in E(G)\}$ are exactly $\{1, 2, \ldots, m\}$. Ringel and Kotzig conjectured in 1964 that every…
We describe those complete linearly ordered topological spaces $X$ which are homogeneous (=CHLOTS). That is, $X$ is order isomorphic with any nonempty open interval in $X$. Using countable tail-like ordinals as indices, we build towers of…
A metric phylogenetic tree relating a collection of taxa induces weighted rooted triples and weighted quartets for all subsets of three and four taxa, respectively. New intertaxon distances are defined that can be calculated from these…