Ranked Trees Research Articles

Biological Properties Influenced by Different Tree Species in Deep Chambal Ravine of Madhya Pradesh, India

An experiment site planted with different multipurpose trees and their impact on soil biological properties were measured and analyzed. Based on overall growth on biological properties showed that preferred ranking tree species in the following order viz., maximum soil microbial biomass carbon, dehydrogenease acticity and soil respiration were recorded under Moring a oleifera, respectively followed by Albizia lebbeck, Azadirachta indica, Acacia nilotica, Dalbergia sissoo, Terminalia arjuna, Millettia pinnata and Gmelina arborea while Justicia adhatoda was recorded minimum, respectively. Tree stand growth significantly improved the soil biological properties. Thus, A. Moringa oleifera and Albizia lebbeck species are most suitable multipurpose hardy tree species for rehabilitation of medium and deep ravines of Chambal ravines in India.

Evaluation of an Image-based Classification Model to Identify Glioma Subtypes Using Arterial Spin Labeling Perfusion MRI On the Publicly Available UCSF Glioma Dataset.

Glioma is acomplex cancer comprising various subtypes and mutations, which may have different metabolic characteristics that can potentially be investigated and identified using perfusion imaging. Therefore, the aim of this work was to use radiomics and machine learning analysis of arterial spin labeling MRI data to automatically differentiate glioma subtypes and mutations. Atotal of 495 Arterial Spin Labeling (ASL) perfusion imaging datasets from the UCSF Glioma database were used in this study. These datasets were segmented to delineate the tumor volume and classified according to tumor grade, pathological diagnosis, and IDH status. Perfusion image data was obtained from a3T MRI scanner using pseudo-continuous ASL. High level texture features were extracted for each ASL dataset using PyRadiomics after tumor volume segmentation and then analyzed using amachine learning framework consisting of ReliefF feature ranking and logistic model tree classification algorithms. The results of the evaluation revealed balanced accuracies for the three endpoints ranging from 55.76% (SD = 4.28, 95% CI: 53.90-57.65) for the tumor grade using 25.4 ± 37.21 features, 62.53% (SD = 2.86, 95% CI: 61.27-63.78) for the mutation status with 23.3 ± 29.17 picked features, and 80.97% (SD = 1.83, 95% CI: 80.17-81.78) for the pathological diagnosis which used 47.3 ± 32.72 selected features. Radiomics and machine learning analysis of ASL perfusion data in glioma patients hold potential for aiding in the diagnosis and treatment of glioma, mainly for discerning glioblastoma from astrocytoma, while performance for tumor grading and mutation status appears limited.

Forward linearised tree pattern matching using tree pattern border array

We define a tree pattern border array as a property of linearised trees analogous to border arrays from the string domain. We consider tree patterns with node wildcards and subtree wildcards. Then we use the tree pattern border array in a new forward tree pattern matching algorithm for ordered ranked trees. The algorithm finds all occurrences of a single linearised tree pattern in a linearised input tree. As with the classical Morris–Pratt algorithm for searching in strings, the tree pattern border array is used to determine shift lengths in the searching phase of the tree pattern matching algorithm. We compare this new algorithm with the best performing previously existing algorithms based on backward linearised tree pattern matching algorithms, (non-)linearised tree pattern matching algorithms using finite tree automata or stringpath matchers. We show that our new algorithm outperforms other tree pattern matching algorithms in single tree pattern matching.

Efficient ensembles of distance‐based label ranking trees

AbstractEnsemble of label ranking trees (LRTs) are currently the state‐of‐the‐art approaches to the label ranking problem. Recently, bagging, boosting, and random forest methods have been proposed, all based on the LRT algorithm, which adapts regression/classification trees to the label classification problem. The LRT algorithm uses theoretically grounded Mallows probability distribution to select the best split when growing the tree, and an EM‐type process to complete the rankings on the training data when they are incomplete. These two steps have proven to be accurate, but require a large computational effort. This article proposes two alternative methods that replace the use of the Mallows distribution with distance‐based criteria to select the best split at each inner node of the tree. Moreover, these distance‐based criteria allow dealing with incomplete rankings natively, so avoiding the completion process. We have carried out an extensive experimental evaluation, which shows that (1) the integration of the two proposed modifications to the LRT algorithm into ensemble methods (bagging and random forest) are an order of magnitude faster than using the original Mallows‐based LRT algorithm; (2) ensembles using the proposed LRT methods are significantly more accurate in the presence of incomplete rankings, while they are at least as accurate in the complete case; and (3) the two modified LRT algorithms are also an order of magnitude faster than the Mallows‐based LRT, while they are at least as accurate as the Mallows‐based LRT on both complete and incomplete rankings.

Ranking trees based on global centrality measures

Trees, or connected graphs with no cycles, are a commonly studied combinatorial family. When many natural metrics on networks are applied to the set of all trees of a fixed order, the extremal values are realized at the star and path graphs. In this paper, we prove inequalities for several global centrality measures, such as global closeness and betweenness centralities, and for graphical Stirling numbers of the first kind that interpolate these extremes. Moreover, we provide two algorithms that allow us to traverse the space of non-isomorphic trees of a fixed order, one towards the star graph of the same order and the other towards the path. Furthermore, we investigate the relationship between these global centrality measures on the one hand and the (n−2)nd Stirling numbers of the first kind for small trees on the other hand, demonstrating a strong association between them, in particular with respect to the hierarchical structures obtained from applying our two interpolating algorithms. Based on our observations from these small trees, we prove general bounds that relate the (n−2)nd Stirling numbers of the first kind of trees of order n to these global centrality measures. Finally, we provide two related approaches to totally order the set of all non-isomorphic trees of fixed order. We show that the totally ordering obtained from one of these approaches is consistent with the hierarchical structure obtained from our two tree interpolation algorithms in addition to being one of the features to use for predicting the (n−2)nd Stirling numbers of the first kind for small trees.

Distribution of External Branch Lengths in Yule Histories

The Yule branching process is a classical model for the random generation of gene tree topologies in population genetics. It generates binary ranked trees -also called histories- with a finite number $n$ of leaves. We study the lengths $\ell_1 > \ell_2 > \cdots > \ell_k > \cdots$ of the external branches of a Yule generated random history of size $n$, where the length of an external branch is defined as the rank of its parent node. When $n \rightarrow \infty$, we show that the random variable $\ell_k$, once rescaled as $\frac{n-\ell_k}{\sqrt{n/2}}$, follows a $\chi$-distribution with $2k$ degrees of freedom, with mean $\mathbb E(\ell_k) \sim n$ and variance $\mathbb V(\ell_k) \sim n \big(k-\frac{\pi k^2}{16^k} \binom{2k}{k}^2\big)$. Our results contribute to the study of the combinatorial features of Yule generated gene trees, in which external branches are associated with singleton mutations affecting individual gene copies.

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.

Cost Automata, Safe Schemes, and Downward Closures

In this work we prove decidability of the model-checking problem for safe recursion schemes against properties defined by alternating B-automata. We then exploit this result to show how to compute downward closures of languages of finite trees recognized by safe recursion schemes. Higher-order recursion schemes are an expressive formalism used to define languages of finite and infinite ranked trees by means of fixed points of lambda terms. They extend regular and context-free grammars, and are equivalent in expressive power to the simply typed λY-calculus and collapsible pushdown automata. Safety in a syntactic restriction which limits their expressive power. The class of alternating B-automata is an extension of alternating parity automata over infinite trees; it enhances them with counting features that can be used to describe boundedness properties.

Can a multistage approach improve individual tree mortality predictions across the complex mixed-species and managed forests of eastern North America?

Tree mortality plays a fundamental role in the dynamics of forest ecosystems, yet it is one of the most difficult phenomena to accurately predict. Various modeling strategies have been developed to improve individual tree mortality predictions. One less explored strategy is the use of a multistage modeling approach. Potential improvements from this approach have remained largely unknown. In this study, we developed a novel multistage approach and compared its performance in individual tree mortality predictions with a more conventional approach using an identical individual tree mortality model formulation. Extensive permanent plot data (n ​= ​9442) covering the Acadian Region of North America and over multiple decades (1965–2014) were used in this study. Our results indicated that the model behavior with the multistage approach better depicted the observed mortality and showed a notable improvement over the conventional approach. The difference between the observed and predicted numbers of dead trees using the multistage approach was much smaller when compared with the conventional approach. In addition, tree survival probabilities predicted by the multistage approach generally were not significantly different from the observations, whereas the conventional approach consistently underestimated mortality across species and overestimated tree survival probabilities over the large range of DBH in the data. The new multistage approach also predictions of zero mortality in individual plots, a result not possible in conventional models. Finally, the new approach was more tolerant of modeling errors because it based estimates on ranked tree mortality rather than error-prone predicted values. Overall, this new multistage approach deserves to be considered and tested in future studies.

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.

A subfunctionalisation model of gene family evolution predicts balanced tree shapes

We consider a subfunctionalisation model of gene family evolution. A family of n genes that perform z functions is represented by an n×z binary matrix Yt where a 1 in the ijth position indicates that gene i can perform function j. Yt evolves according to a continuous time Markov chain (CTMC) that represents the processes of gene duplication, coding region loss and regulatory region loss with the restriction that each function is protected by selection, meaning that each column in the matrix must contain at least one 1.We generate gene trees based on the CTMC {Yt,t⩾0}. We analyse the long-run behaviour of the model and specify the conditions where we expect gene trees to continue to grow and where we expect them to have a stable number of genes. We show that different choices of rate parameters for the processes of duplication and loss lead to different tree shapes as measured by two common tree-shape statistics: the β-statistic for measuring tree balance and the γ-statistic for assessing diversification rate. We use an extension of β that is estimated from sets of trees. This extension is less biased compared to using the average β value for individual trees.When the rate of gene duplication dominates the rate of gene loss, the process is unstable and the distribution of tree shapes is close to following the uniform ranked tree shape (URT) distribution. However, when the process is stable, gene trees are predicted to have positive values of β indicating balanced trees and negative values of γ indicating that diversification occurs more towards the root of the tree. The results of our analyses suggest that comparing the tree-shape statistics of empirical gene-trees to the predictions presented here will provide a test of the subfunctionalisation model.

Ensemble learning for the partial label ranking problem

The partial label ranking problem is a general interpretation of the preference learning scenario known as the label ranking problem, the goal of which is to learn preference classifiers able to predict a complete ranking with ties over the finite set of labels of the class variable. In this paper, we use unsupervised discretization techniques (equal‐frequency and equal‐width binning) to heuristically select the threshold for the numerical features in the algorithms based on induction of decision trees (partial label ranking trees algorithm). Moreover, we adapt the most well‐known averaging (bootstrap aggregating and random forests) and boosting (adaptive boosting) ensemble methods to the partial label ranking problem, in order to improve the robustness of the built classifiers. We compare the proposed methods with the nearest neighbors‐based algorithm (instance based partial label ranking) over the standard benchmark datasets, showing that our versions of the ensemble methods are superior in terms of accuracy. Furthermore, they are affordable in terms of computational efficiency.

Enumeration of binary trees compatible with a perfect phylogeny

Evolutionary models used for describing molecular sequence variation suppose that at a non-recombining genomic segment, sequences share ancestry that can be represented as a genealogy—a rooted, binary, timed tree, with tips corresponding to individual sequences. Under the infinitely-many-sites mutation model, mutations are randomly superimposed along the branches of the genealogy, so that every mutation occurs at a chromosomal site that has not previously mutated; if a mutation occurs at an interior branch, then all individuals descending from that branch carry the mutation. The implication is that observed patterns of molecular variation from this model impose combinatorial constraints on the hidden state space of genealogies. In particular, observed molecular variation can be represented in the form of a perfect phylogeny, a tree structure that fully encodes the mutational differences among sequences. For a sample of n sequences, a perfect phylogeny might not possess n distinct leaves, and hence might be compatible with many possible binary tree structures that could describe the evolutionary relationships among the n sequences. Here, we investigate enumerative properties of the set of binary ranked and unranked tree shapes that are compatible with a perfect phylogeny, and hence, the binary ranked and unranked tree shapes conditioned on an observed pattern of mutations under the infinitely-many-sites mutation model. We provide a recursive enumeration of these shapes. We consider both perfect phylogenies that can be represented as binary and those that are multifurcating. The results have implications for computational aspects of the statistical inference of evolutionary parameters that underlie sets of molecular sequences.

Effect of dominant shade tree species on selected soil physicochemical properties and coffee production in Sayyo district, western Ethiopia

The current study aimed to assess the preference of farmers and their perception of shade tree species on coffee production and to evaluate selected soil physicochemical properties under the canopies of Acacia abyssinica and Albizia gummifera shade tree species and the influence of these trees on coffee production in comparison with unshaded fields. Two-stage sampling technique was used to select respondents for this study. Four circular plots having 3, 5, and 7 m radius and 17 m were laid out to sample soil and coffee parameters. The top six ranked tree species preferred by farmers were, Acacia abyssinica Albizia gummifera, Croton macrostachyus, Cordia Africana, Millettia ferruginea, and Erythrina abyssinica. The soil organic matter, total nitrogen, available phosphorous, exchangeable potassium, and soil pH were higher underneath shade trees than in open areas. Soils tested under shade trees and in open areas were texturally similar, (sandy clay loam). This pattern of textural similarity indicated that they have been derived from the same parent material. There was greater coffee yield, coffee weight, and coffee shrub density under the tree canopy than in open fields. Generally, Coffee production parameters and coffee yield under the canopies of both shade tree species and selected soil physicochemical properties showed relatively higher than those of unshaded zones. Therefore, integrating Acacia abyssinica and Albizia gummifera shade trees in coffee production systems is very important in maintaining soil fertility and thereby increasing the yield of organic coffee.

Discrete coalescent trees

In many phylogenetic applications, such as cancer and virus evolution, time trees, evolutionary histories where speciation events are timed, are inferred. Of particular interest are clock-like trees, where all leaves are sampled at the same time and have equal distance to the root. One popular approach to model clock-like trees is coalescent theory, which is used in various tree inference software packages. Methodologically, phylogenetic inference methods require a tree space over which the inference is performed, and the geometry of this space plays an important role in statistical and computational aspects of tree inference algorithms. It has recently been shown that coalescent tree spaces possess a unique geometry, different from that of classical phylogenetic tree spaces. Here we introduce and study a space of discrete coalescent trees. They assume that time is discrete, which is natural in many computational applications. This tree space is a generalisation of the previously studied ranked nearest neighbour interchange space, and is built upon tree-rearrangement operations. We generalise existing results about ranked trees, including an algorithm for computing distances in polynomial time, and in particular provide new results for both the space of discrete coalescent trees and the space of ranked trees. We establish several geometrical properties of these spaces and show how these properties impact various algorithms used in phylogenetic analyses. Our tree space is a discretisation of a previously introduced time tree space, called t-space, and hence our results can be used to approximate solutions to various open problems in t-space.

Higher-order Recursion Schemes and Collapsible Pushdown Automata: Logical Properties

This article studies the logical properties of a very general class of infinite ranked trees, namely, those generated by higher-order recursion schemes. We consider, for both monadic second-order logic and modal -calculus, three main problems: model-checking, logical reflection (a.k.a. global model-checking, that asks for a finite description of the set of elements for which a formula holds), and selection (that asks, if exists, for some finite description of a set of elements for which an MSO formula with a second-order free variable holds). For each of these problems, we provide an effective solution. This is obtained, thanks to a known connection between higher-order recursion schemes and collapsible pushdown automata and on previous work regarding parity games played on transition graphs of collapsible pushdown automata.

Heuristics for unrooted, unranked, and ranked anomaly zones under birth-death models

Species trees that can generate a nonmatching gene tree topology that is more probable than the topology matching the species tree are said to be in an anomaly zone. We introduce some heuristic approaches to infer whether species trees are in anomaly zones when it is difficult or impossible to compute the entire distribution of gene tree topologies. Here, probabilities of unrooted, unranked, and ranked gene tree topologies under the multispecies coalescent are used. A ranked tree can be viewed as an unranked tree with a temporal ordering of its internal nodes. Overall, considering probabilities of unrooted or unranked gene tree topologies within one nearest neighbor interchange from the species tree topology is a reasonable heuristic to infer the existence of anomalous unrooted or unranked gene trees, respectively. We investigated a test proposed by Linkem et al. (2016) which classifies a species tree as being in an unranked anomaly zone if there is a subset of four taxa in an unranked anomaly zone. We find this test to have high true positive rates, but it can also have high false positive rates. For ranked trees, because at least one of the most probable ranked gene tree topologies must have the same unranked topology as the species tree, we propose to use only those ranked gene trees that have topologies that match the unranked species tree topology. We find that the probability that the species tree is in unrooted and unranked anomaly zones tends to increase with the speciation rate, and the probability of all three types of anomaly zones increases rapidly with the number of taxa. We find that probabilities that species trees are in an anomaly zone can be quite high for moderately high speciation rates.

Streaming ranked-tree-to-string transducers

Streaming tree transducers with single-use restriction (STTsurs) were introduced by Alur and D'Antoni as an analyzable, executable, and expressive model for transforming unranked ordered trees in a single pass. The equivalence problem of STTsurs is decidable because their class is as expressive as the class of MSO-definable tree transformations. In this paper, we present streaming ranked-tree-to-string transducers (SRTSTs), based on STTsurs: SRTSTs are released from the single-use restriction while their input and output are restricted to ranked trees and strings, respectively. We show that the expressiveness of SRTSTs coincides with that of deterministic top-down tree transducers with regular look-ahead (yDTRs), whose equivalence problem is known to be decidable. Our proof is done by constructing equivalent transducers in both directions.

Computing nearest neighbour interchange distances between ranked phylogenetic trees

Many popular algorithms for searching the space of leaf-labelled (phylogenetic) trees are based on tree rearrangement operations. Under any such operation, the problem is reduced to searching a graph where vertices are trees and (undirected) edges are given by pairs of trees connected by one rearrangement operation (sometimes called a move). Most popular are the classical nearest neighbour interchange, subtree prune and regraft, and tree bisection and reconnection moves. The problem of computing distances, however, is {mathbf {N}}{mathbf {P}}-hard in each of these graphs, making tree inference and comparison algorithms challenging to design in practice. Although ranked phylogenetic trees are one of the central objects of interest in applications such as cancer research, immunology, and epidemiology, the computational complexity of the shortest path problem for these trees remained unsolved for decades. In this paper, we settle this problem for the ranked nearest neighbour interchange operation by establishing that the complexity depends on the weight difference between the two types of tree rearrangements (rank moves and edge moves), and varies from quadratic, which is the lowest possible complexity for this problem, to {mathbf {N}}{mathbf {P}}-hard, which is the highest. In particular, our result provides the first example of a phylogenetic tree rearrangement operation for which shortest paths, and hence the distance, can be computed efficiently. Specifically, our algorithm scales to trees with tens of thousands of leaves (and likely hundreds of thousands if implemented efficiently).

Optimal Trees Selection for Classification via Out-of-Bag Assessment and Sub-Bagging

The effect of training data size on machine learning methods has been well investigated over the past two decades. The predictive performance of tree based machine learning methods, in general, improves with a decreasing rate as the size of training data increases. We investigate this in optimal trees ensemble (OTE) where the method fails to learn from some of the training observations due to internal validation. Modified tree selection methods are thus proposed for OTE to cater for the loss of training observations in internal validation. In the first method, corresponding out-of-bag (OOB) observations are used in both individual and collective performance assessment for each tree. Trees are ranked based on their individual performance on the OOB observations. A certain number of top ranked trees is selected and starting from the most accurate tree, subsequent trees are added one by one and their impact is recorded by using the OOB observations left out from the bootstrap sample taken for the tree being added. A tree is selected if it improves predictive accuracy of the ensemble. In the second approach, trees are grown on random subsets, taken without replacement-known as sub-bagging, of the training data instead of bootstrap samples (taken with replacement). The remaining observations from each sample are used in both individual and collective assessments for each corresponding tree similar to the first method. Analysis on 21 benchmark datasets and simulations studies show improved performance of the modified methods in comparison to OTE and other state-of-the-art methods.