Centrifugal-order distributions in binary topological trees

Statistical properties of topological binary trees are studied on the basis of the distribution of segments in relation to centrifugal order. Special attention is paid to the mean of this distribution in a tree as it will be used as a measure of tree topology. It will be shown how the expectation of the mean centrifugal order depends both on the size of the tree and on the mode of growth in the context of modelling the growth of tree structures. Observed trees can be characterized by their mean orders and procedures are described to find the growth mode that optimally corresponds to these data. The variance structure of the mean-order measure appears to be a crucial factor in these fitting procedures. Examples indicate that mean-order analysis is an accurate alternative to partition analysis that is based on the partitioning of segments over sub-tree pairs at branching points.     

A scaling law for the effects of architecture and allometry on tree vibration modes suggests a biological tuning to modal compartmentalization

Wind is a major ecological factor for plants and a major economical factor for forestry. Mechanical analyses have revealed that the multimodal dynamic behavior of trees is central to wind-tree interactions. Moreover, the trunk and branches influence dynamic modes, both in frequency and location. Because of the complexity of tree architecture, finite element models (FEMs) have been used to analyze such dynamics. However, these models require detailed geometric and architectural data and are tree-specific-two major restraints for their use in most ecological or biological studies. In this work, closed-form scaling laws for modal characteristics were derived from the dimensional analysis of idealized fractal trees that sketched the major architectural and allometrical regularities of real trees. These scaling laws were compared to three-dimensional FEM modal analyses of two completely digitized trees with maximal architectural contrast. Despite their simplifying hypotheses, the models explained most of the spatiotemporal characteristics of modes that involved the trunk and branches, especially for sympodial trees. These scaling laws reduce the tree to (1) a fundamental frequency and (2) one architectural and three biometrical parameters. They also give quantitative insights into the possible biological control of wind excitability of trees through architecture and allometries.     

Gene tree distributions under the coalescent process

Under the coalescent model for population divergence, lineage sorting can cause considerable variability in gene trees generated from any given species tree. In this paper, we derive a method for computing the distribution of gene tree topologies given a bifurcating species tree for trees with an arbitrary number of taxa in the case that there is one gene sampled per species. Applications for gene tree distributions include determining exact probabilities of topological equivalence between gene trees and species trees and inferring species trees from multiple datasets. In addition, we examine the shapes of gene tree distributions and their sensitivity to changes in branch lengths, species tree shape, and tree size. The method for computing gene tree distributions is implemented in the computer program COAL.     

Architecture and size relations: an essay on the apple (Malus x domestica, Rosaceae) tree

The influence of tree size independent of age on some architectural features (annual shoot length, lateral branching, flowering) was investigated on 4-yr-old apple (Malus × domestica) trees either own-rooted or grafted on the dwarfing rootstock M.9, giving rise to large and small trees, respectively. Tree size significantly affected the length of the first annual shoot of bottom branches with a lesser effect on the subsequent annual shoots of the same branches and on branches situated higher in the tree canopy. The linear regression parameters, i.e., slopes and intercepts, between annual shoot length and number of growing laterals were affected by the genotype and, depending on genotype, by tree size. Flowering was generally lower, delayed, and more irregular on large trees compared to small trees, with on average similar ranking of genotypes regardless of tree size. This study provides evidence for a specific effect of tree size, as affected by the root system, on architectural development of the apple tree regardless of the genotype. From an architectural viewpoint, the dwarfing mechanism could be interpreted as a faster physiological aging essentially related to the reduction in length of the first annual shoot of bottom branches and the high flowering on this shoot.     

Maximum tree: a consistent estimator of the species tree

We propose a model based approach to use multiple gene trees to estimate the species tree. The coalescent process requires that gene divergences occur earlier than species divergences when there is any polymorphism in the ancestral species. Under this scenario, speciation times are restricted to be smaller than the corresponding gene split times. The maximum tree (MT) is the tree with the largest possible speciation times in the space of species trees restricted by available gene trees. If all populations have the same population size, the MT is the maximum likelihood estimate of the species tree. It can be shown the MT is a consistent estimator of the species tree even when the MT is built upon the estimates of the true gene trees if the gene tree estimates are statistically consistent. The MT converges in probability to the true species tree at an exponential rate.     

Comparative height crown allometry and mechanical design in 22 tree species of Kuala Belalong rainforest, Brunei, Borneo

In rainforests, trunk size, strength, crown position, and geometry of a tree affect light interception and the likelihood of mechanical failure. Allometric relationships of tree diameter, wood density, and crown architecture vs. height are described for a diverse range of rainforest trees in Brunei, northern Borneo. The understory species follow a geometric model in their diameter-height relationship (slope, β = 1.08), while the stress-elasticity models prevail (β = 1.27-1.61) for the midcanopy and canopy/emergent species. These relationships changed with ontogeny, especially for the understory species. Within species, the tree stability safety factor (SSF) and relative crown width decreased exponentially with increasing tree height. These trends failed to emerge in across-species comparisons and were reversed at a common (low) height. Across species, the relative crown depth decreased with maximum potential height and was indistinguishable at a common (low) height. Crown architectural traits influence SSF more than structural property of wood density. These findings emphasize the importance of applying a common reference size in comparative studies and suggest that forest trees (especially the understory group) may adapt to low light by having deeper rather than wider crowns due to an efficient distribution and geometry of their foliage.     

Remodelling during development of the Purkinje cell dendritic tree in the mouse

The development of the Purkinje cells in normal C57 mice was studied from 7-100 d post natum . The growth of the dendritic trees was analysed both metrically and topologically using the method of vertex analysis (Berry & Flinn 1983 a). Granule and PUrkinje cell counts were made so that Purkinje cell segment production could be correlated with the number of parallel fibres deposited. Both topological and metrical results indicate that from 7 to 30 d post natum the Purkinje cell dendritic trees expand massively; accounting for 87% of total segment elaboration, reaching their lateral boundaries by 12-15 d post natum and then advancing towards the pial surface. Continued lateral expansion is constrained by the proximity of dendrites from neighbouring trees. Growth proceeds upwards through the neuropil as a front of prolific random terminal branching with inhibitory forces acting at the edges of the growth corridor and behind the growth front to prevent overlapping of dendrites. By 30 d post natum all boundaries are reached and the size of the dendritic field is fixed. Trees averaged 711.2 segments +/- 21.45 with a mean distance from root to terminal segment of 133.5 +/- 2.9 micrometers. The Va/Vb vertex ratios and the levels of trichotomy during this period indicate that branching patterns deviate from pure random terminal additions in a dichotomous tree. There is opportunity for non-random growth at the areas of inhibitory action. Beyond 30 d post natum remodelling occurs within the arbor which involves segment loss in the subpial region (orders above 16) and segment elaboration within the tree (orders 8-16) causing increased density of dendrites and overlapping of segments. The frequencies of segments and terminals are restored to symmetrical distributions through the orders of the trees from the skewed distributions associated with the frontal advance in earlier growth. During remodelling the Va/Vb vertex ratios and percentage of trichotomous nodes are consistent with growth through dichotomous random terminal branching. Path lengths of 8 micrometers between each order are seen as regular increments throughout entire trees at 100 d post natum . The final tree produced is indistinguishable from a network grown entirely by random terminal dichotomous branching with some 6% trichotomy and a Va/Vb vertex ratios of 0.92. Granule cell number within the granular layer increases rapidly up to 15 d post natum after which cell death causes a decrease to stable levels beyond 30 d post natum .(ABSTRACT TRUNCATED AT 400 WORDS)     

A mathematical model to describe the dynamic response of a spruce tree to the wind

 A mathematical, computer-based, dynamic sway model of a Sitka spruce (Picea sitchensis) tree was developed and tested against measurements of the movement of a tree within a forest. The model tree was divided into segments each with a stiffness, mass and damping parameter. Equations were formulated to describe the response of every segment which together form a system of coupled differential equations. These were solved with the aid of matrices and from the resulting modes, the transfer function of the tree was found and used to calculate the movement of the tree in the wind. Comparison of the modelled movement of a tree in response to the measured wind speed above a forest canopy gave good agreement with the measured movement of the top of the tree but less satisfactory agreement close to the base. The comparison also pointed to the complexity of tree response to the wind and inadequacies in the model. In particular, the branches need to be treated as coupled cantilevers attached to the stem rather than simply as masses lumped together. Received: 18 February 1997 / Accepted: 16 December 1997     

Topological properties of binary trees grown with order-dependent branching probabilities

This paper describes a growth model for binary topological trees. The model defines the branching probability of all segments in the tree. The branching probability of a segment is formulated as a function of two variables, one indicating its type (intermediate or terminal), the other representing its order, i.e. the topological distance to the root segment. The function is determined by two parameters, namely the ratio of branching probabilities of intermediate and terminal segments and the strength of the order dependency, implemented in an exponential form. Expressions are derived for the calculation of symmetry properties of the partitions and it is indicated which part of the parameter domain results in predominantly symmetrical trees.     

Response of tree biomass and wood litter to disturbance in a Central Amazon forest

We developed an individual-based stochastic-empirical model to simulate the carbon dynamics of live and dead trees in a Central Amazon forest near Manaus, Brazil. The model is based on analyses of extensive field studies carried out on permanent forest inventory plots, and syntheses of published studies. New analyses included: (1) growth suppression of small trees, (2) maximum size (trunk base diameter) for 220 tree species, (3) the relationship between growth rate and wood density, and (4) the growth response of surviving trees to catastrophic mortality (from logging). The model simulates a forest inventory plot, and tracks recruitment, growth, and mortality of live trees, decomposition of dead trees (coarse litter), and how these processes vary with changing environmental conditions. Model predictions were tested against aggregated field data, and also compared with independent measurements including maximum tree age and coarse litter standing stocks. Spatial analyses demonstrated that a plot size of ~10 ha was required to accurately measure wood (live and dead) carbon balance. With the model accurately predicting relevant pools and fluxes, a number of model experiments were performed to predict forest carbon balance response to perturbations including: (1) increased productivity due to CO₂ fertilization, (2) a single semi-catastrophic (10%) mortality event, (3) increased recruitment and mortality (turnover) rates, and (4) the combined effects of increased turnover, increased tree growth rates, and decreased mean wood density of new recruits. Results demonstrated that carbon accumulation over the past few decades observed on tropical forest inventory plots (~0.5 Mg C ha^–1 year^–1) is not likely caused by CO₂ fertilization. A maximum 25% increase in woody tissue productivity with a doubling of atmospheric CO₂ only resulted in an accumulation rate of 0.05 Mg C ha^–1 year^–1 for the period 1980–2020 for a Central Amazon forest, or an order of magnitude less than observed on the inventory plots. In contrast, model parameterization based on extensive data from a logging experiment demonstrated a rapid increase in tree growth following disturbance, which could be misinterpreted as carbon sequestration if changes in coarse litter stocks were not considered. Combined results demonstrated that predictions of changes in forest carbon balance during the twenty-first century are highly dependent on assumptions of tree response to various perturbations, and underscores the importance of a close coupling of model and field investigations.     

Host tree architecture mediates the effect of predators on herbivore survival

Abstract.  1. Vegetation structural complexity is an important factor influencing ecological interactions between different trophic levels. In order to investigate relationships between the architecture of trees, the presence of arthropod predators, and survival and parasitism of the autumnal moth Epirrita autumnata Borkhausen, two sets of experiments were conducted.
2. In one experiment, the architectural complexity of mountain birch was manipulated to separate the effects of plant structure and age. In the other experiment the trees were left intact, but chosen to represent varying degrees of natural complexity. Young autumnal moth larvae were placed on the trees and their survival was monitored during the larval period.
3. The larvae survived longer in more complex trees if predation by ants was prevented with a glue ring, whereas in control trees smaller canopy size improved survival times in one experiment. The density of ants observed in the trees was not affected by canopy size but spider density was higher on smaller trees. The effect of canopy structure on larval parasitism was weak; larger canopy size decreased parasitism only in one year. Until the fourth instar the larvae travelled shorter distances in trees with reduced branchiness than in trees with reduced foliage or control treatments. Canopy structure manipulation by pruning did not alter the quality of leaves as food for larvae.
4. The effect of canopy structure on herbivore survival may depend on natural enemy abundance and foraging strategy. In complex canopies herbivores are probably better able to escape predation by ambushing spiders but not by actively searching ants.     

Statistical properties of bootstrap estimation of phylogenetic variability from nucleotide sequences. I. Four taxa with a molecular clock.

The statistical properties of sample estimation and bootstrap estimation of phylogenetic variability from a sample of nucleotide sequences are studied by using model trees of three taxa with an outgroup and by assuming a constant rate of nucleotide substitution. The maximum-parsimony method of tree reconstruction is used. An analytic formula is derived for estimating the sequence length that is required if P, the probability of obtaining the true tree from the sampled sequences, is to be equal to or higher than a given value. Bootstrap estimation is formulated as a two-step sampling procedure: (1) sampling of sequences from the evolutionary process and (2) resampling of the original sequence sample. The probability that a bootstrap resampling of an original sequence sample will support the true tree is found to depend on the model tree, the sequence length, and the probability that a randomly chosen nucleotide site is an informative site. When a trifurcating tree is used as the model tree, the probability that one of the three bifurcating trees will appear in > or = 95% of the bootstrap replicates is < 5%, even if the number of bootstrap replicates is only 50; therefore, the probability of accepting an erroneous tree as the true tree is < 5% if that tree appears in > or = 95% of the bootstrap replicates and if more than 50 bootstrap replications are conducted. However, if a particular bifurcating tree is observed in, say, < 75% of the bootstrap replicates, then it cannot be claimed to be better than the trifurcating tree even if > or = 1,000 bootstrap replications are conducted. When a bifurcating tree is used as the model tree, the bootstrap approach tends to overestimate P when the sequences are very short, but it tends to underestimate that probability when the sequences are long. Moreover, simulation results show that, if a tree is accepted as the true tree only if it has appeared in > or = 95% of the bootstrap replicates, then the probability of failing to accept any bifurcating tree can be as large as 58% even when P = 95%, i.e., even when 95% of the samples from the evolutionary process will support the true tree. Thus, if the rate-constancy assumption holds, bootstrapping is a conservative approach for estimating the reliability of an inferred phylogeny for four taxa.     

A polynomial time algorithm for calculating the probability of a ranked gene tree given a species tree

ABSTRACT: BACKGROUND: The ancestries of genes form gene trees which do not necessarily have the same topology as the species tree due to incomplete lineage sorting. Available algorithms determining the probability of a gene tree given a species tree require exponential computational runtime. RESULTS: In this paper, we provide a polynomial time algorithm to calculate the probability of a ranked gene tree topology for a given species tree, where a ranked tree topology is a tree topology with the internal vertices being ordered. The probability of a gene tree topology can thus be calculated in polynomial time if the number of orderings of the internal vertices is a polynomial number. However, the complexity of calculating the probability of a gene tree topology with an exponential number of rankings for a given species tree remains unknown. CONCLUSIONS: Polynomial algorithms for calculating ranked gene tree probabilities may become useful in developing methodology to infer species trees based on a collection of gene trees, leading to a more accurate reconstruction of ancestral species relationships.     

Theoretical relationship between the optimal models of the vascular tree

Two models of optimal branching structure of the vascular tree are compared. Murray’s minimum work model derived from minimum energy loss due to flow and volume in the duct system is proved to be included as a mathematical group in the authors’ model defined by the minimum volume under determinant pressure, flow and position at the terminals. The problem about heterotypical trees which are identical at the terminal conditions but different in the topological order of branch combinations are discussed, applying the results of analyses on the equivalent duct of uniform terminal pressure trees. It is proved that the minimum work tree has the least energy loss compared with its heterotypical minimum volume trees and is a better model of branching structure of the vascular tree.     

Local-scale drivers of tree survival in a temperate forest

Tree survival plays a central role in forest ecosystems. Although many factors such as tree size, abiotic and biotic neighborhoods have been proposed as being important in explaining patterns of tree survival, their contributions are still subject to debate. We used generalized linear mixed models to examine the relative importance of tree size, local abiotic conditions and the density and identity of neighbors on tree survival in an old-growth temperate forest in northeastern China at three levels (community, guild and species). Tree size and both abiotic and biotic neighborhood variables influenced tree survival under current forest conditions, but their relative importance varied dramatically within and among the community, guild and species levels. Of the variables tested, tree size was typically the most important predictor of tree survival, followed by biotic and then abiotic variables. The effect of tree size on survival varied from strongly positive for small trees (1-20 cm dbh) and medium trees (20-40 cm dbh), to slightly negative for large trees (>40 cm dbh). Among the biotic factors, we found strong evidence for negative density and frequency dependence in this temperate forest, as indicated by negative effects of both total basal area of neighbors and the frequency of conspecific neighbors. Among the abiotic factors tested, soil nutrients tended to be more important in affecting tree survival than topographic variables. Abiotic factors generally influenced survival for species with relatively high abundance, for individuals in smaller size classes and for shade-tolerant species. Our study demonstrates that the relative importance of variables driving patterns of tree survival differs greatly among size classes, species guilds and abundance classes in temperate forest, which can further understanding of forest dynamics and offer important insights into forest management.     

Scelestial: Fast and accurate single-cell lineage tree inference based on a Steiner tree approximation algorithm

Single-cell genome sequencing provides a highly granular view of biological systems but is affected by high error rates, allelic amplification bias, and uneven genome coverage. This creates a need for data-specific computational methods, for purposes such as for cell lineage tree inference. The objective of cell lineage tree reconstruction is to infer the evolutionary process that generated a set of observed cell genomes. Lineage trees may enable a better understanding of tumor formation and growth, as well as of organ development for healthy body cells. We describe a method, Scelestial, for lineage tree reconstruction from single-cell data, which is based on an approximation algorithm for the Steiner tree problem and is a generalization of the neighbor-joining method. We adapt the algorithm to efficiently select a limited subset of potential sequences as internal nodes, in the presence of missing values, and to minimize cost by lineage tree-based missing value imputation. In a comparison against seven state-of-the-art single-cell lineage tree reconstruction algorithms—BitPhylogeny, OncoNEM, SCITE, SiFit, SASC, SCIPhI, and SiCloneFit—on simulated and real single-cell tumor samples, Scelestial performed best at reconstructing trees in terms of accuracy and run time. Scelestial has been implemented in C++. It is also available as an R package named RScelestial.     

Effect of age on the distribution of oil in Eastern redcedar tree segments

Eastern redcedar is widespread in the US and produces significant amount of biomass. Open-grown trees invade abandoned fields and compete with valuable forage species in pastures and rangelands. Value-added product development from redcedar is vital for management of eastern redcedar. Cedarwood oil is a valuable component which can be used for further value-added product development. This study examined the effect of age on the distribution of oil in redcedar tree segments. Trunks of eastern redcedar (Juniperus virginiana L.) trees at different stages of growth (26-63 years old) were divided into three sections (top, center and lower). Each section was fractionated separately into bark, heartwood and sapwood segments. Heartwood and sapwood samples from each tree section were analyzed for oil content and composition. A hydrodistillation method was used for oil extraction. Volatile components of tree segments were examined by using a Gas Chromatograph-headspace analysis technique. The heartwood of eastern redcedar contained significantly higher oil than sapwood. Older trees had more oil in the heartwood than younger trees. Both redcedar bark and leaves contained significantly lower oil content than the cedarwood. There were also significant differences in the oil composition of bark, leaves and wood fractions. Cedarwood oil extraction may benefit from prior separation of tree segments prior to oil extraction. However, the economic feasibility of separation prior to an extraction process needs to be further studied. Required extra capital investment and operating costs need to be examined, as well as whether sapwood is worth processing.     

Genetic determinism of the vegetative and reproductive traits in an F1 olive tree progeny

The agronomic performance of fruit trees is significantly influenced by tree internal organization. Introducing architectural traits in breeding programs could thus lead to select new varieties with a regular bearing and lower input demand in order to reduce training and environmental costs. However, an interaction between tree ontogeny and genetic factors is expected. In this study, we investigated the genetic determinism of architectural traits in the olive tree, accounting for tree development over 5 years until first flowering occurrence. We studied an F1 progeny issued from a cross between two contrasted genotypes, ‘Olivière’ and ‘Arbequina’. Tree architecture was decomposed in quantitative traits, related to (1) growth and branching, (2) first flowering and fruiting. Models, including the year of growth, branching order and genotype effects, were built with variance function and covariance structure when necessary. After a model selection, broad sense heritabilities were calculated. During the first 3 years, both the mean values of vegetative traits and genetic factor significance depended on the shoot within-tree position. Dependencies between consecutive years were revealed for traits related to whole tree form. Whole tree form variables showed medium to high broad sense heritability values, whereas reproductive traits were highly heritable. This study demonstrates the existence of ontogenic trends in the olive tree, which result in traits heritable only at the tree periphery. A phenotyping strategy adapted to its architectural characteristics and a list of relevant traits, such as maximal internode length, is proposed. Transgressive effects suggest that genetic progress could be performed in future selection programs.     

Pair hidden Markov models on tree structures

MOTIVATION: Computationally identifying non-coding RNA regions on the genome has much scope for investigation and is essentially harder than gene-finding problems for protein-coding regions. Since comparative sequence analysis is effective for non-coding RNA detection, efficient computational methods are expected for structural alignments of RNA sequences. On the other hand, Hidden Markov Models (HMMs) have played important roles for modeling and analysing biological sequences. Especially, the concept of Pair HMMs (PHMMs) have been examined extensively as mathematical models for alignments and gene finding. RESULTS: We propose the pair HMMs on tree structures (PHMMTSs), which is an extension of PHMMs defined on alignments of trees and provides a unifying framework and an automata-theoretic model for alignments of trees, structural alignments and pair stochastic context-free grammars. By structural alignment, we mean a pairwise alignment to align an unfolded RNA sequence into an RNA sequence of known secondary structure. First, we extend the notion of PHMMs defined on alignments of 'linear' sequences to pair stochastic tree automata, called PHMMTSs, defined on alignments of 'trees'. The PHMMTSs provide various types of alignments of trees such as affine-gap alignments of trees and an automata-theoretic model for alignment of trees. Second, based on the observation that a secondary structure of RNA can be represented by a tree, we apply PHMMTSs to the problem of structural alignments of RNAs. We modify PHMMTSs so that it takes as input a pair of a 'linear' sequence and a 'tree' representing a secondary structure of RNA to produce a structural alignment. Further, the PHMMTSs with input of a pair of two linear sequences is mathematically equal to the pair stochastic context-free grammars. We demonstrate some computational experiments to show the effectiveness of our method for structural alignments, and discuss a complexity issue of PHMMTSs.     

Comparing tree shapes: beyond symmetry

This paper describes two types of problems related to tree shapes, as well as algorithms that can be used to solve these problems. The first problem is that of comparing the similarity of the unlabelled shapes instead of merely their degree of balance, in a manner analogous to that routinely used to compare topologies for labelled trees. There are possible practical applications for this comparison, such as determining, based on tree shape similarity alone, whether the taxa in two phylogenies are likely to have a correspondence (e.g. hosts and parasites with high specificity). It is shown that tree balance is insufficient for this task and that standard measures of topological difference (Robinson–Foulds distances, SPR distances or retention indices of the matrices representing the trees, MRPs) can be easily adapted to the problem. The second type of problem is to determine whether taxa of uncertain matching unique to two different phylogenies could correspond to each other (e.g. the same species in larvae and adults of metamorphic animals, fossils known from different body parts). This second problem can be solved by either relabelling taxa in such a way that the number of consensus nodes is maximized, or relabelling taxa in such a way that the sum of the number of steps in the MRP of each tree mapped onto the other is minimum.