Related papers: Counting tanglegrams with species
We study the class of graphs known as k-trees through the lens of Joyal's theory of combinatorial species (and an equivariant extension known as '$\Gamma$-species' which incorporates data about 'structural' group actions). This culminates…
A string-like compact data structure for unlabelled rooted trees is given using 2n bits.
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 show how an image can, in principle, be described by the tangles of the graph of its pixels. The tangle-tree theorem provides a nested set of separations that efficiently distinguish all the distinguishable tangles in a graph. This…
This extended abstract is dedicated to the analysis of the height of non-plane unlabelled rooted binary trees. The height of such a tree chosen uniformly among those of size $n$ is proved to have a limiting theta distribution, both in a…
Complex systems are usually represented as an intricate set of relations between their components forming a complex graph or network. The understanding of their functioning and emergent properties are strongly related to their structural…
Full binary trees naturally represent commutative non-associative products. There are many important examples of these products: finite-precision floating-point addition and NAND gates, among others. Balance in such a tree is highly…
P\'olya trees are rooted, unlabeled trees on $n$ vertices. This paper gives an efficient, new way to generate P\'olya trees. This allows comparing typical unlabeled and labeled tree statistics and comparing asymptotic theorems with…
A closed-form formula is derived for the number of occurrences of matches of a multiset of patterns among all ordered (plane-planted) trees with a given number of edges. A pattern looks like a tree, with internal nodes and leaves, but also…
Diagrammatic notation has become a ubiquitous computational tool; early examples include Penrose's graphical notation for tensor calculus, Feynman's diagrams for perturbative quantum field theory, and Cvitanovic's birdtracks for Lie…
We give a short and direct proof of a remarkable identity that arises in the enumeration of labeled trees with respect to their indegree sequence, where all edges are oriented from the vertex with lower label towards the vertex with higher…
Given two messages - as linear sequences of letters, it is immediate to determine whether one can be transformed into the other by simple substitution cipher of the letters. On the other hand, if the letters are carried as labels on nodes…
Lajos Takacs gave a somewhat formidable alternating sum formula for the number of forests of unrooted trees on $n$ labeled vertices. Here we use a weight-reversing involution on suitable tree configurations to give a combinatorial…
An increasing 1,2-tree is a labeled graph formed by starting with a vertex and then repeatedly attaching a leaf to a vertex or a triangle to an edge, the labeling of the vertices corresponding to the order in which the vertices are added.…
We consider finite trees with edges labeled by letters on a finite alphabet $\varSigma$. Each pair of nodes defines a unique labeled path whose trace is a word of the free monoid $\varSigma^*$. The set of all such words defines the language…
The occurrence and the distribution of patterns of trees associated to natural numbers are investigated. Bounds from above and below are proven for certain natural quantities.
We prove a new formula for the generating function of multitype Cayley trees counted according to their degree distribution. Using this formula we recover and extend several enumerative results about trees. In particular, we extend some…
We solve a class of optimization problems for (phylogenetic) $X$-trees or their shapes. These problems have recently appeared in different contexts, e.g. in the context of the impact of tree shapes on the size of TBR neighborhoods, but so…
In agglomerative hierarchical clustering, pair-group methods suffer from a problem of non-uniqueness when two or more distances between different clusters coincide during the amalgamation process. The traditional approach for solving this…
Tree-like tableaux are objects in bijection with alternative or permutation tableaux. They have been the subject of a fruitful combinatorial study for the past few years. In the present work, we define and study a new subclass of tree-like…