Unlabeled Trees Research Articles

Enumeration of Rooted Binary Unlabeled Galled Trees

Rooted binary galled trees generalize rooted binary trees to allow a restricted class of cycles, known as galls. We build upon the Wedderburn-Etherington enumeration of rooted binary unlabeled trees with n leaves to enumerate rooted binary unlabeled galled trees with n leaves, also enumerating rooted binary unlabeled galled trees with n leaves and g galls, 0⩽g⩽⌊n-12⌋\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0 \\leqslant g \\leqslant \\lfloor \\frac{n-1}{2} \\rfloor $$\\end{document}. The enumerations rely on a recursive decomposition that considers subtrees descended from the nodes of a gall, adopting a restriction on galls that amounts to considering only the rooted binary normal unlabeled galled trees in our enumeration. We write an implicit expression for the generating function encoding the numbers of trees for all n. We show that the number of rooted binary unlabeled galled trees grows with 0.0779(4.8230n)n-32\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0.0779(4.8230^n)n^{-\\frac{3}{2}}$$\\end{document}, exceeding the growth 0.3188(2.4833n)n-32\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0.3188(2.4833^n)n^{-\\frac{3}{2}}$$\\end{document} of the number of rooted binary unlabeled trees without galls. However, the growth of the number of galled trees with only one gall has the same exponential order 2.4833 as the number with no galls, exceeding it only in the subexponential term, 0.3910n12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0.3910n^{\\frac{1}{2}}$$\\end{document} compared to 0.3188n-32\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0.3188n^{-\\frac{3}{2}}$$\\end{document}. For a fixed number of leaves n, the number of galls g that produces the largest number of rooted binary unlabeled galled trees lies intermediate between the minimum of g=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g=0$$\\end{document} and the maximum of g=⌊n-12⌋\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g=\\lfloor \\frac{n-1}{2} \\rfloor $$\\end{document}. We discuss implications in mathematical phylogenetics.

Connectivity Maintenance through Unlabeled Spanning Tree Matching

A key function of mobile networks is the ability to dynamically reshape itself to any desired geometry. Lacking absolute position awareness, agents often rely on distance-limited inter-agent spatial measurements to maintain state awareness. Methods of formation control must therefore ensure a minimal level of persistent pairwise measurement feedback throughout transition, giving rise to the classic connectivity maintenance problem. To address this problem, we propose a method of structure-preserving assignment, matching agents to desired positions such that persistent global connectivity is naturally and automatically satisfied under smooth transition. Compared to other approaches, this complementary technique reduces reliance on aggressive or costly mid-flight formation control protocols. The technique is shown to scale and even improve with network size.

Early share of 15N-labelled fertilizer between trees and crop in young temperate alley-cropping system

Nitrogen (N) partitioning between trees and crops in alley-cropping systems (AC) may increase the N fertilizer use, but more investigation is needed to elucidate it for young trees. The aim of this study was to explore the share of N between crops and three tree species after a few years of tree establishment in AC. Using 15N labeling (10 atm% as 15NH4,15NO3 form), we traced the fate of N fertilizer in wheat associated with four-year-old trees at Ramecourt alley-cropping experimental site. The 15N abundance of tree leaves and crop whole-plant was compared between samples collected in AC, in sole-crop control (CC) and in unlabeled and unfertilized pure-forest control plots (FC). The wheat δ15N increased from 0.7 ± 0.6‰ to 994 ± 474‰ before and 12 days after labeling and remained stable until harvest. For natural abundance, no significant difference in wheat δ15N was observed between AC and CC. The same result was observed for labeled treatment. The percentage of wheat N derived from fertilizer was on average 27 ± 17% and no effect of type of system or associated tree species was observed. Tree foliar δ15N was multiplied by 1.5, 1.8 and 4 for willow, hornbeam and wild cherry respectively compared to unlabeled trees in AC. The percentage of tree foliar N derived from fertilizer was on average 0.04 ± 0.04% and no significant difference was observed between tree species. The wheat associated with trees allocated significantly more tracer N to ears than wheat in monocrop. From this study, we succeeded in quantifying the beginning of tree/crop sharing of N fertilizer using labeling method and evidenced tree species effect. Young trees in AC contributed to valorize the N fertilizer without significant effect on global wheat uptake and favored its valorization in wheat grain.

Bijections between the multifurcating unlabeled rooted trees and the positive integers

Colijn and Plazzotta (2018) [1] described a bijective scheme for associating the unlabeled bifurcating rooted trees with the positive integers. In mathematical and biological applications of unlabeled rooted trees, however, nodes of rooted trees are sometimes multifurcating rather than bifurcating. Building on the bijection between the unlabeled bifurcating rooted trees and the positive integers, we describe bijective schemes for associating the unlabeled multifurcating rooted trees with the positive integers. We devise bijections with the positive integers for a set of trees in which each non-leaf node has exactly k child nodes, and for a set of trees in which each non-leaf node has at most k child nodes. The calculations make use of Macaulay's binomial expansion formula. The generalization to multifurcating trees can assist with the use of unlabeled trees for applications in evolutionary biology, such as the measurement of phylogenetic patterns of genetic lineages in pathogens.

Statistical summaries of unlabelled evolutionary trees.

Rooted and ranked phylogenetic trees are mathematical objects that are useful in modelling hierarchical data and evolutionary relationships with applications to many fields such as evolutionary biology and genetic epidemiology. Bayesian phylogenetic inference usually explores the posterior distribution of trees via Markov chain Monte Carlo methods. However, assessing uncertainty and summarizing distributions remains challenging for these types of structures. While labelled phylogenetic trees have been extensively studied, relatively less literature exists for unlabelled trees that are increasingly useful, for example when one seeks to summarize samples of trees obtained with different methods, or from different samples and environments, and wishes to assess the stability and generalizability of these summaries. In our paper, we exploit recently proposed distance metrics of unlabelled ranked binary trees and unlabelled ranked genealogies, or trees equipped with branch lengths, to define the Fréchet mean, variance and interquartile sets as summaries of these tree distributions. We provide an efficient combinatorial optimization algorithm for computing the Fréchet mean of a sample or of distributions on unlabelled ranked tree shapes and unlabelled ranked genealogies. We show the applicability of our summary statistics for studying popular tree distributions and for comparing the SARS-CoV-2 evolutionary trees across different locations during the COVID-19 epidemic in 2020. Our current implementations are publicly available at https://github.com/RSamyak/fmatrix.

Two results about the Sackin and Colless indices for phylogenetic trees and their shapes.

The Sackin and Colless indices are two widely-used metrics for measuring the balance of trees and for testing evolutionary models in phylogenetics. This short paper contributes two results about the Sackin and Colless indices of trees. One result is the asymptotic analysis of the expected Sackin and Colless indices of tree shapes (which are full binary rooted unlabelled trees) under the uniform model where tree shapes are sampled with equal probability. Another is a short direct proof of the closed formula for the expected Sackin index of phylogenetic trees (which are full binary rooted trees with leaves being labelled with taxa) under the uniform model.

An adjacent-swap Markov chain on coalescent trees

AbstractThe standard coalescent is widely used in evolutionary biology and population genetics to model the ancestral history of a sample of molecular sequences as a rooted and ranked binary tree. In this paper we present a representation of the space of ranked trees as a space of constrained ordered matched pairs. We use this representation to define ergodic Markov chains on labeled and unlabeled ranked tree shapes analogously to transposition chains on the space of permutations. We show that an adjacent-swap chain on labeled and unlabeled ranked tree shapes has a mixing time at least of order $n^3$ , and at most of order $n^{4}$ . Bayesian inference methods rely on Markov chain Monte Carlo methods on the space of trees. Thus it is important to define good Markov chains which are easy to simulate and for which rates of convergence can be studied.

Root finding algorithms and persistence of Jordan centrality in growing random trees

We consider models of growing random trees {Tf(n):n≥1} with model dynamics driven by an attachment function f:Z+→R+. At each stage a new vertex enters the system and connects to a vertex v in the current tree with probability proportional to f(degree(v)). The main goal of this study is to understand the performance of root finding algorithms. A large body of work (e.g., Random Structures Algorithms 50 (2017) 158–172; IEEE Trans. Netw. Sci. Eng. 4 (2017) 1–12; Random Structures Algorithms 52 (2018) 136–157) has emerged in the last few years in using techniques based on the Jordan centrality measure (J. Reine Angew. Math. 70 (1869) 185–190) and its variants to develop root finding algorithms. Given an unlabeled unrooted tree, one computes the Jordan centrality for each vertex in the tree and for a fixed budget K outputs the optimal K vertices (as measured by Jordan centrality). Under general conditions on the attachment function f, we derive necessary and sufficient bounds on the budget K(ε) in order to recover the root with probability at least 1−ε. For canonical examples such as linear preferential attachment and uniform attachment, these general results give matching upper and lower bounds for the budget. We also prove persistence of the optimal K Jordan centers for any K, that is, the existence of an almost surely finite random time n∗ such that for n≥n∗ the identity of the K-optimal Jordan centers in {Tf(n):n≥n∗} does not change, thus describing robustness properties of this measure. Key technical ingredients in the proofs of independent interest include sufficient conditions for the existence of exponential moments for limits of (appropriately normalized) continuous time branching processes within which the models {Tf(n):n≥1} can be embedded, as well as rates of convergence results to these limits.

How Does Sample Labeling and Distribution Affect the Accuracy and Efficiency of a Deep Learning Model for Individual Tree-Crown Detection and Delineation

Monitoring and assessing vegetation using deep learning approaches has shown promise in forestry applications. Sample labeling to represent forest complexity is the main limitation for deep learning approaches for remote sensing vegetation classification applications, and few studies have focused on the impact of sample labeling methods on model performance and model training efficiency. This study is the first-of-its-kind that uses Mask region-based convolutional neural networks (Mask R-CNN) to evaluate the influence of sample labeling methods (including sample size and sample distribution) on individual tree-crown detection and delineation. A flight was conducted over a plantation with Fokienia hodginsii as the main tree species using a Phantom4-Multispectral (P4M) to obtain UAV imagery, and a total of 2061 manually and accurately delineated tree crowns were used for training and validating (1689) and testing (372). First, the model performance of three pre-trained backbones (ResNet-34, ResNet-50, and ResNet-101) was evaluated. Second, random deleting and clumped deleting methods were used to repeatedly delete 10% from the original sample set to reduce the training and validation set, to simulate two different sample distributions (the random sample set and the clumped sample set). Both RGB image and Multi-band images derived from UAV flights were used to evaluate model performance. Each model’s average per-epoch training time was calculated to evaluate the model training efficiency. The results showed that ResNet-50 yielded a more robust network than ResNet-34 and ResNet-101 when the same parameters were used for Mask R-CNN. The sample size determined the influence of sample labeling methods on the model performance. Random sample labeling had lower requirements for sample size compared to clumped sample labeling, and unlabeled trees in random sample labeling had no impact on model training. Additionally, the model with clumped samples provides a shorter average per-epoch training time than the model with random samples. This study demonstrates that random sample labeling can greatly reduce the requirement of sample size, and it is not necessary to accurately label each sample in the image during the sample labeling process.

Analyzing Phylogenetic Trees with a Tree Lattice Coordinate System and a Graph Polynomial.

Phylogenetic trees are a central tool in many areas of life science and medicine. They demonstrate evolutionary patterns among species, genes, and patterns of ancestry among sets of individuals. The tree shapes and branch lengths of phylogenetic trees encode evolutionary and epidemiological information. To extract information from tree shapes and branch lengths, representation and comparison methods for phylogenetic trees are needed. Representing and comparing tree shapes and branch lengths of phylogenetic trees are challenging, for a tree shape is unlabeled and can be displayed in numerous different forms, and branch lengths of a tree shape are specific to edges whose positions vary with respect to the displayed forms of the tree shape. In this article, we introduce representation and comparison methods for rooted unlabeled phylogenetic trees based on a tree lattice that serves as a coordinate system for rooted binary trees with branch lengths and a graph polynomial that fully characterizes tree shapes. We show that the introduced tree representations and metrics provide distance-based likelihood-free methods for tree clustering, parameter estimation, and model selection and apply the methods to analyze phylogenies reconstructed from virus sequences. [Graph polynomial; likelihood-free inference; phylogenetics; tree lattice; tree metrics.].

Broadcasting on random recursive trees

We study the broadcasting problem when the underlying tree is a random recursive tree. The root of the tree has a random bit value assigned. Every other vertex has the same bit value as its parent with probability 1−q and the opposite value with probability q, where q∈[0,1]. The broadcasting problem consists in estimating the value of the root bit upon observing the unlabeled tree, together with the bit value associated with every vertex. In a more difficult version of the problem, the unlabeled tree is observed but only the bit values of the leaves are observed. When the underlying tree is a uniform random recursive tree, in both variants of the problem we characterize the values of q for which the optimal reconstruction method has a probability of error bounded away from 1/2. We also show that the probability of error is bounded by a constant times q. Two simple reconstruction rules are analyzed in detail. One of them is the simple majority vote, the other is the bit value of the centroid of the tree. Most results are extended to linear preferential attachment trees as well.

Lossless Compression of Semi-Ordered Trees

In this paper, authors are interested in the problem of lossless compression of unlabeled semi-ordered trees. Semi-ordered trees are a class of trees that present an order between some sibling while some other sibling are unordered. They offer a wide possibility of applications especially for the representation of plants architecture. Authors show that these trees present remarkable compression properties covering those of ordered and unordered trees. To illustrate this approach, authors apply these notions to a particular class of semi-ordered trees which is the most studied branching structure particularly for a botanical motivation, namely axial trees.

The power of Tarski's relation algebra on trees

Fragments of Tarski's relation algebra form the basis of many versatile graph and tree query languages including the regular path queries, XPath, and SPARQL. Surprisingly, however, a systematic study of the relative expressive power of relation algebra fragments on trees has not yet been undertaken. In this work, we perform such a systematic study. Our approach is to start from a basic fragment which only allows composition and union. We then study how the expressive power of the query language changes if we add diversity, converse, projections, coprojections, intersection, and/or difference, both for path queries and Boolean queries. For path queries on labeled trees, we found that adding intersection and difference yields more expressive power for some fragments, while adding one of the other operators always yields more expressive power. For Boolean queries on labeled trees, we obtain a similar picture for the relative expressive power, except for a few fragments where adding converse or projection yields no more expressive power. Additionally, we also studied querying unlabeled trees, for which we have found several redundancies. One challenging problem remains open, however, for both path and Boolean queries: does adding difference yield more expressive power to fragments containing at least diversity, coprojections, and intersection?

Extremal inverse eigenvalue problem for irreducible acyclic matrices

In this paper, we study the inverse eigenvalue problem of constructing symmetric matrices whose graph is a tree, i.e. of constructing irreducible acyclic matrices from given eigendata consisting of the smallest and largest eigenvalues of their leading principal submatrices. We solve the problem completely for the case when the given eigenvalues are distinct and the subgraph of the tree induced by remains connected for . The result generalizes the results known for matrices described by some specific trees. The proofs of the main results are constructive and so provide an algorithm for computing the actual entries of the required matrix. Numerical solutions of the problem are then obtained with SCILAB by feeding the eigendata and plugging in the adjacency matrix as inputs. We also make an attempt to find the number of solutions for several special families of unlabelled trees.

Peripheral structures in unlabelled trees and the accumulation of subgenomes in the evolution of polyploids

Many angiosperms have undergone some series of polyploidization events over the course of their evolutionary history. In these genomes, especially those resulting from multiple autopolyploidization, it may be relatively easy to recognize all the ξ sets of n homeologous chromosomes, but it is much harder, if not impossible, to partition these chromosomes into n subgenomes, each representing one distinct genomic component of ξ chromosomes making up the original polyploid. Thus, if we wish to infer the polyploidization history of the genome, we could make use of all the gene trees inferred from the genes in one set of homeologous chromosomes to construct a consensus tree, but there is no evident way of combining the trees from the ξ different sets, because we have no labelling of the chromosomes that is known to be consistent across these sets. We suggest here that lacking a consistent leaf-labelling, the topological structure of the trees may display sufficient resemblance so that a higher level consensus could be revealing of evolutionary history. This would be especially true of the peripheral structures of the tree, likely representing events that occurred more recently and have thus been less obscured by subsequent evolutionary processes.Here, we present a statistical test to assess whether the subgenomes in a polyploid genome could have been added one at a time. The null hypothesis is that the accumulation of chromosomes follows a stochastic process in which transition from one generation to the next is through randomly choosing an edge, and then subdividing this edge in order to link the new internal vertex to a new external vertex.We analyze the probability distributions of a number of peripheral tree substructures, namely leaf- or terminal-pairs, triples and quadruples, arising from this stochastic process, in terms of some exact recurrences. We propose some conjectures regarding the asymptotic behaviours of these distributions. Applying our analysis to a sugarcane genome, we demonstrate that it is unlikely that the accumulation of subgenomes has occurred one at a time.

Non-crossing Trees, Quadrangular Dissections, Ternary Trees, and Duality-Preserving Bijections

Using the theory of Properly Embedded Graphs developed in an earlier work we define an involutory duality on the set of labeled non-crossing trees that lifts the obvious duality in the set of unlabeled non-crossing trees. The set of non-crossing trees is a free ternary magma with one generator and this duality is an instance of a duality that is defined in any such magma. Any two free ternary magmas with one generator are isomorphic via a unique isomorphism that we call the structural bijection. Besides the set of non-crossing trees we also consider as free ternary magmas with one generator the set of ternary trees, the set of quadrangular dissections, and the set of flagged Perfectly Chain Decomposed Ditrees, and we give topological and/or combinatorial interpretations of the structural bijections between them. In particular the bijection from the set of quadrangular dissections to the set of non-crossing trees seems to be new. Further we give explicit formulas for the number of self-dual labeled and unlabeled non-crossing trees and the set of quadrangular dissections up to rotations and up to rotations and reflections.

Protection numbers in simply generated trees and Pólya trees

We determine the limit of the expected value and the variance of the protection number of the root in simply generated trees, in P?lya trees, and in unlabelled non-plane binary trees, when the number of vertices tends to infinity. Moreover, we compute expectation and variance of the protection number of a randomly chosen vertex in all those tree classes. We obtain exact formulas as sum representations, where the obtained sums are rapidly converging thus allowing an efficient numerical computation of high accuracy.

Encoding and decoding algorithms for unlabeled trees

Trees considered in this article, in which at least two receivers emanate from each internal vertex, are found in the works by E. Schroder, R. Stanley, O.V. Kuzmin, etc. Encoding and decoding algorithms for unlabeled planar rooted trees with a given number of end vertices, root receivers, and a sequence of degrees of internal tree vertices during depth-first search are built. Encoding is done in non-decreasing tuples, by assigning labels to the internal vertices of the tree. The proposed algorithms made it possible to prove the existence of a one-to-one correspondence between the studied set of trees and the set of non-decreasing tuples. To find the cardinality of a set of tuples, we used an approach based on the generalized Pascal pyramid.

A tree distinguishing polynomial

We define a bivariate polynomial for unlabeled rooted trees and show that the polynomial of an unlabeled rooted tree T is the generating function of a class of subtrees of T. We prove that the polynomial is a complete isomorphism invariant for unlabeled rooted trees. Then, we generalize the polynomial to unlabeled unrooted trees and we show that the generalized polynomial is a complete isomorphism invariant for unlabeled unrooted trees.

Tree shape-based approaches for the comparative study of cophylogeny.

Cophylogeny is the congruence of phylogenetic relationships between two different groups of organisms due to their long‐term interaction. We investigated the use of tree shape distance measures to quantify the degree of cophylogeny. We implemented a reverse‐time simulation model of pathogen phylogenies within a fixed host tree, given cospeciation probability, host switching, and pathogen speciation rates. We used this model to evaluate 18 distance measures between host and pathogen trees including two kernel distances that we developed for labeled and unlabeled trees, which use branch lengths and accommodate different size trees. Finally, we used these measures to revisit published cophylogenetic studies, where authors described the observed associations as representing a high or low degree of cophylogeny. Our simulations demonstrated that some measures are more informative than others with respect to specific coevolution parameters especially when these did not assume extreme values. For real datasets, trees’ associations projection revealed clustering of high concordance studies suggesting that investigators are describing it in a consistent way. Our results support the hypothesis that measures can be useful for quantifying cophylogeny. This motivates their usage in the field of coevolution and supports the development of simulation‐based methods, i.e., approximate Bayesian computation, to estimate the underlying coevolutionary parameters.