Related papers: The Proportion of Trees that are Linear
A subtree can be induced in a natural way by a subset of leaves of a rooted tree. We study the number of nonisomorphic such subtrees induced by leaves (leaf-induced subtrees) of a rooted tree with no vertex of outdegree 1 (topological…
An explicit perturbative solution to all orders is given for a general class of nonlinear differential equations. This solution is written as a sum indexed by rooted trees and uses the Green function of a linearization of the equations. The…
A central limit theorem for binary tree is numerically examined. Two types of central limit theorem for higher-order branches are formulated. A topological structure of a binary tree is expressed by a binary sequence, and the…
We establish limit theorems that describe the asymptotic local and global geometric behaviour of random enriched trees considered up to symmetry. We apply these general results to random unlabelled weighted rooted graphs and uniform random…
We consider planar rooted random trees whose distribution is even for fixed height $h$ and size $N$ and whose height dependence is given by a power function $h^\alpha$. Defining the total weight for such trees of fixed size to be $Z_N$, a…
We present some exact expressions for the number of paths of a given length in a perfect $m$-ary tree. We first count the paths in perfect rooted $m$-ary trees and then use the results to determine the number of paths in perfect unrooted…
Species trees represent the historical divergences of populations or species, while gene trees trace the ancestry of individual gene copies sampled within those populations. In cases involving rapid speciation, gene trees with topologies…
When considering the number of subtrees of trees, the extremal structures which maximize this number among binary trees and trees with a given maximum degree lead to some interesting facts that correlate to other graphical indices in…
We answer three questions posed by Bubeck and Linial on the limit densities of subtrees in trees. We prove there exist positive $\varepsilon_1$ and $\varepsilon_2$ such that every tree that is neither a path nor a star has inducibility at…
Let $p(m)$ (respectively, $q(m)$) be the maximum number $k$ such that any tree with $m$ edges can be transformed by contracting edges (respectively, by removing vertices) into a caterpillar with $k$ edges. We derive closed-form expressions…
We consider an infinite sequence of rooted trees naturally emerging in a number-theoretical context. We advance some ideas on its structure by discussing some elementary properties. Some of those properties are shown to be related to…
Regular tree grammars and regular path expressions constitute core constructs widely used in programming languages and type systems. Nevertheless, there has been little research so far on frameworks for reasoning about path expressions…
We give general theorems on asymptotic normality for additive functionals of random tries generated by a sequence of independent strings. These theorems are applied to show asymptotic normality of the distribution of random fringe trees in…
We introduce a solvable model of randomly growing systems consisting of many independent subunits. Scaling relations and growth rate distributions in the limit of infinite subunits are analysed theoretically. Various types of scaling…
We consider growing random recursive trees in random environment, in which at each step a new vertex is attached (by an edge of a random length) to an existing tree vertex according to a probability distribution that assigns the tree…
Quasi-trees generalize trees in that the unique "path" between two nodes may be infinite and have any countable order type. They are used to define the rank-width of a countable graph in such a way that it is equal to the least upper-bound…
We investigate the structure of trees that have minimal algebraic connectivity among all trees with a given degree sequence. We show that such trees are caterpillars and that the vertex degrees are non-decreasing on every path on…
Trees without vertices of degree $2$ are sometimes named topological trees. In this work, we bring forward the study of the inducibility of (rooted) topological trees with a given number of leaves. The inducibility of a topological tree $S$…
We find a simple, closed formula for the proportion of vertices which are $k$-protected in all unlabeled rooted plane trees on $n$ vertices. We also find that, as $n$ goes to infinity, the average rank of a random vertex in a tree of size…
While the notion of arboricity of a graph is well-known in graph theory, very few results are dedicated to the minimal number of trees covering the edges of a graph, called the tree number of a graph.