Morphometric analysis of root systems: application of the technique and influence of soil fertility on root system development in two herbaceous species

Abstract. A method for describing root systems based on geomorphological techniques developed for river systems is described. Root systems, in common with other natural branching structures (rivers, bronchioles, trees), appear to obey Morton's Law of Branching: there is a constant ratio, the bifurcation or branching ratio, R_b, between the number of branches of a given order, N_u , and that of the next order. N_u+1 , In experiments where Poa annua , and Rumex cripus , were grown at two levels of fertility, the first-order roots (the youngest members in this system) were generally unresponsive to fertility, and differences in the root systems were largely the result of changes in the second-order roots, those formed at the junction of two first-order roots. These differences were reflected in the branching ratio, R_b Although it is possible to explain these results by a stochastic model of branch development, the R_b values for roots are higher than for other natural branching structures, and higher than the random model predicts. It is possible that a model based on optimum exploration of space may be more appropriate and provide a key to the factors governing root branching patterns.     

Tree asymmetry—A sensitive and practical measure for binary topological trees

The topological structure of a binary tree is characterized by a measure called tree asymmetry, defined as the mean value of the asymmetry of its partitions. The statistical properties of this tree-asymmetry measure have been studied using a growth model for binary trees. The tree-asymmetry measure appears to be sensitive for topological differences and the tree-asymmetry expectation for the growth model that we used appears to be almost independent of the size of the trees. These properties and the simple definition make the measure suitable for practical use, for instance for characterizing, comparing and interpreting sets of branching patterns. Examples are given of the analysis of three sets of neuronal branching patterns. It is shown that the variance in tree-asymmetry values for these observed branching patterns corresponds perfectly with the variance predicted by the used growth model.     

Arterial branching within the confines of fractal L-system formalism

Parametric Lindenmayer systems (L-systems) are formulated to generate branching tree structures that can incorporate the physiological laws of arterial branching. By construction, the generated trees are de facto fractal structures, and with appropriate choice of parameters, they can be made to exhibit some of the branching patterns of arterial trees, particularly those with a preponderant value of the asymmetry ratio. The question of whether arterial trees in general have these fractal characteristics is examined by comparison of pattern with vasculature from the cardiovascular system. The results suggest that parametric L-systems can be used to produce fractal tree structures but not with the variability in branching parameters observed in arterial trees. These parameters include the asymmetry ratio, the area ratio, branch diameters, and branching angles. The key issue is that the source of variability in these parameters is not known and, hence, it cannot be accurately reproduced in a model. L-systems with a random choice of parameters can be made to mimic some of the observed variability, but the legitimacy of that choice is not clear.     

Quantifying the degree of self-nestedness of trees: application to the structural analysis of plants

In this paper, we are interested in the problem of approximating trees by trees with a particular self-nested structure. Self-nested trees are such that all their subtrees of a given height are isomorphic. We show that these trees present remarkable compression properties, with high compression rates. In order to measure how far a tree is from being a self-nested tree, we then study how to quantify the degree of self-nestedness of any tree. For this, we define a measure of the self-nestedness of a tree by constructing a self-nested tree that minimizes the distance of the original tree to the set of self-nested trees that embed the initial tree. We show that this measure can be computed in polynomial time and depict the corresponding algorithm. The distance to this nearest embedding self-nested tree (NEST) is then used to define compression coefficients that reflect the compressibility of a tree. To illustrate this approach, we then apply these notions to the analysis of plant branching structures. Based on a database of simulated theoretical plants in which different levels of noise have been introduced, we evaluate the method and show that the NESTs of such branching structures restore partly or completely the original, noiseless, branching structures. The whole approach is then applied to the analysis of a real plant (a rice panicle) whose topological structure was completely measured. We show that the NEST of this plant may be interpreted in biological terms and may be used to reveal important aspects of the plant growth.     

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.     

Predicting root biomass from branching patterns of Douglas-fir root systems

There are many examples of branching networks in nature, such as tree crowns, river systems, arteries and lungs. These networks have often been described as being self-similar, or following scale-invariant branching rules, and this property has been used to derive several scaling laws. In this paper we model root systems of Douglas-fir ( Pseudotsuga menziesii var. glauca (Beissn.) Franco) as branching networks following several simple branching rules. Our objective is to establish a relationship between trunk diameter and root biomass. We explore the effect of the self-similar branching assumption on this relationship. Using data collected from a mature stand in British Columbia, we find that branching asymmetry and the rate of root taper change with root size, thereby violating the assumption of self-similarity. However, the data are in general agreement with Leonardo da Vinci's area-preserving branching hypothesis. We use the field data to parameterize two models, one assuming self-similar branching and a second incorporating the measured size dependencies of branching parameters. The two models differ by only a small amount (≈8%) in their predictions. For both models, the predicted relationship between trunk diameter and root biomass is in good concordance with previously published empirical data. We conclude that the assumption of self-similar branching, although violated by the data, nevertheless provides a useful tool for predicting the allometric relationship between trunk diameter and root biomass. Finally, we use our models to show that the geometric properties of individual bifurcations fundamentally change the root biomass cost of different root topologies.     

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.     

BRANCH GEOMETRY AND EFFECTIVE LEAF AREA: A STUDY OF TERMINAUA-BRANCHING PATTERN. 1. THEORETICAL TREES

Single lateral branches and branch tiers of Terminalia catappa L. are simulated and drawn by computer. Leaf clusters on the branches are approximated by discs, and the effective leaf areas are determined by use of Dirichlet domains. Theoretical optimal branching angles which produce the maximum effective leaf area are obtained from simulations. Symmetrical and asymmetrical branching angles are contrasted; the latter characterize real trees. Varying leaf disc radius and ratio of branch-unit lengths affects optimal branching angles, as does the symmetry of a tier of five branches. Leaf area indices for individual branches and branch tiers are given for all simulations. The number of branches in a tier has a major effect on leaf area index and effective leaf area. The theoretical optimal branching angles of many simulations are very close to the values observed in real trees of T. catappa. We conclude that the observed branching angles and number of branches in a tier of this species optimize light interception within constraints of a fixed pattern of branching, one that is widespread among tropical trees.     

Constructing rooted supertrees using distances

Suppose that a family of rooted phylogenetic trees T _i with different sets X _i of leaves is given. A supertree for the family is a single rooted tree T whose leaf set is the union of all the X _i , such that the branching information in T corresponds to the branching information in all the trees T _i . This paper proposes a polynomial-time method BUILD-WITH-DISTANCES that makes essential use of distance information provided by the trees T _i to construct a rooted tree S ₀. When a supertree also containing the distance information exists, then S ₀ is a supertree. The supertree S ₀ often shows increased resolution over the trees found by methods that utilize only the topology of the input trees. When no supertree exists because the input trees are incompatible, several variants of the method are described which still produce trees with provable properties.     

Emergence of matched airway and vascular trees from fractal rules

The bronchial, arterial, and venous trees of the lung are complex interwoven structures. Their geometries are created during fetal development through common processes of branching morphogenesis. Insights from fractal geometry suggest that these extensively arborizing trees may be created through simple recursive rules. Mathematical models of Turing have demonstrated how only a few proteins could interact to direct this branching morphogenesis. Development of the airway and vascular trees could, therefore, be considered an example of emergent behavior as complex structures are created from the interaction of only a few processes. However, unlike inanimate emergent structures, the geometries of the airway and vascular trees are highly stereotyped. This review will integrate the concepts of emergence, fractals, and evolution to demonstrate how the complex branching geometries of the airway and vascular trees are ideally suited for gas exchange in the lung. The review will also speculate on how the heterogeneity of blood flow and ventilation created by the vascular and airway trees is overcome through their coordinated construction during fetal development.     

The Algebra of the General Markov Model on Phylogenetic Trees and Networks

It is known that the Kimura 3ST model of sequence evolution on phylogenetic trees can be extended quite naturally to arbitrary split systems. However, this extension relies heavily on mathematical peculiarities of the associated Hadamard transformation, and providing an analogous augmentation of the general Markov model has thus far been elusive. In this paper, we rectify this shortcoming by showing how to extend the general Markov model on trees to include incompatible edges; and even further to more general network models. This is achieved by exploring the algebra of the generators of the continuous-time Markov chain together with the “splitting” operator that generates the branching process on phylogenetic trees. For simplicity, we proceed by discussing the two state case and then show that our results are easily extended to more states with little complication. Intriguingly, upon restriction of the two state general Markov model to the parameter space of the binary symmetric model, our extension is indistinguishable from the Hadamard approach only on trees; as soon as any incompatible splits are introduced the two approaches give rise to differing probability distributions with disparate structure. Through exploration of a simple example, we give an argument that our extension to more general networks has desirable properties that the previous approaches do not share. In particular, our construction allows for convergent evolution of previously divergent lineages; a property that is of significant interest for biological applications.     

Branching structure in arborescent animals: Models of relative growth

Many invertebrate animals belonging to diverse phyla grow as regularly branching structures with the general appearance of miniature trees. If it is assumed that regularity of branching implies regularity in growth, models can be mathematically derived to depict growth of such a structure as a set of changing morphologic properties. Modes of growth, branching properties, and growth models can be expected to differ markedly from one major taxonomic group to another. Nevertheless, these properties can furnish a useful basis for comparing adaptive morphologies and underlying mechanical designs not only among arborescent animals, but with arborescent plants as well.Branching structures of some cheilostome bryozoans with rigidly erect, arborescent growth habits are inferred to result from continuous growth at steadily increasing numbers of growing tips through a process of repeated bifurcation and lengthening. In a model of continuous growth, the pattern by which the number of growing tips increases can be shown to be a generalized mathematical series, of which the Fibonacci series and a geometric series are two special cases. The quantities which determine the series can be calculated from measurable properties of the branching structure: lengths of paired branch portions ending in growing tips (relative growth ratio), lengths of paired branch portions between bifurcations (mean link length and link-length ratio), and numbers of branch portions belonging to different orders (branching ratio). Data for eight species of cheilostome bryozoans indicate, with high levels of confidence, that measurable branching properties and the models of relative growth inferred from them are species-specific. This specificity and a tendency to adhere to characteristic values of branching properties during growth are apparently direct expressions of internal control in these bryozoans.     

Distribution of end-points of a branching network with decaying branch length

The geometry of the human bronchial tree has been described as a network formed by successive dichotomous branching with constant branching angles and geometrically decaying branch lengths. Models having these properties and with randomly distributed branching planes are constructed. The distribution of the end points of the model networks are described by computing the variance of the distributions in the direction of the axis of the network and in the transverse directions. It is found that, for a given decay ratio, there is a branching angle for which the volume filled by the end points is a maximum. The advantages of the network with the decay ratio and branching angle of the human bronchial tree are discussed.     

Biological solutions to transport network design

Transport networks are vital components of multicellular organisms, distributing nutrients and removing waste products. Animal and plant transport systems are branching trees whose architecture is linked to universal scaling laws in these organisms. In contrast, many fungi form reticulated mycelia via the branching and fusion of thread-like hyphae that continuously adapt to the environment. Fungal networks have evolved to explore and exploit a patchy environment, rather than ramify through a three-dimensional organism. However, there has been no explicit analysis of the network structures formed, their dynamic behaviour nor how either impact on their ecological function. Using the woodland saprotroph Phanerochaete velutina, we show that fungal networks can display both high transport capacity and robustness to damage. These properties are enhanced as the network grows, while the relative cost of building the network decreases. Thus, mycelia achieve the seemingly competing goals of efficient transport and robustness, with decreasing relative investment, by selective reinforcement and recycling of transport pathways. Fungal networks demonstrate that indeterminate, decentralized systems can yield highly adaptive networks. Understanding how these relatively simple organisms have found effective transport networks through a process of natural selection may inform the design of man-made networks.     

Development of a mathematical method for classifying and comparing tree architecture using parameters from a topological model of a trifurcating botanical tree

This paper describes a model for the topological mapping of trifurcating botanical trees. The model was based on a system of modular units that represented the interconnectivity of shoot meristems (terminal segments) and internodes (internal segments) within whole plant canopies, organized with increasing centrifugal ordering. The model was capable of describing the dynamics of plant growth as expressed by changes in topological parameters over time. Preliminary calculations for experimental trees indicated that the model represents growth in a biologically sound manner. Methods are described for the calculation of the architecture parameters size, size-complexity, structural complexity, and tree asymmetry index (TAI). Parameter calculations were based on the mathematical principles developed for the classification of bifurcating dendrite trees, and were designed to both extract structural information, and to enable statistical comparison between trees of different size. Parameters were mathematically adjusted for trifurcation, and appeared to be able to represent quantitatively the architectural properties of tree structures. In addition to the calculation of the TAI for trifurcating trees, new methods were developed to enable comparisons to be made of the architectural complexity of trifurcating trees of differing size. These were based on the principle of the pair-wise comparison of the mean centrifugal order number (MCON) with respect to segments against highest order number. We argue and illustrate that this principle can be more informative than that of pair-wise comparison of the MCON against tree degree (topological size). Further improvements to this method were made by examining branching points (vertices) rather than segments (links) to calculate the MCON.     

Computer simulation of the geometry of the human bronchial tree

Some morphological features of the human bronchial tree were simulated by computergenerated trees. The trees were ordered by the methods of Horsfield and Strahler. Delta, the difference between the Horsfield orders of the two branches at a bifurcation, was determined by pseudorandom numbers generated according to a distribution of probabilities defined on input. By trial and error a distribution was found which resulted in trees being generated with average Strahler order branching ratios of 2.82, similar to a real bronchial tree. Branching angles and length ratio could also be defined on input. By varying these input parameters it was found that the form of the tree was quite sensitive to them, and that by a suitable choice the intrasegmental part of the bronchial tree could be simulated. It is concluded that branching ratio, length ratio, mean branching angles and distribution of delta are controlled within tight limits in the bronchial tree, and this may support the concept of optimal design.     

Cuttrees in the topological analysis of branching patterns

In this paper the effects of the occurrence of cut trees in the topological analysis of branching patterns have been studied. It is assumed that branches are removed at random from the trees. We prove that, for both the segmental and terminal growth models, the probability distributions of the cut trees are identical to those of complete trees.     

Why are there so many threat displays?

The morphology of branching trees in general, and of the bronchial tree in particular, can be described in terms of three parameters, the diameter, length and branching ratios. These are the factors by which mean diameter and mean length increase in successive orders towards the trachea, and by which the number of branches increases in successive orders away from the trachea. Orders of branching are counted from the periphery towards the trachea, according to the method of Strahler. A model of from two to nine orders, and of constant total length and volume, was used to investigate the effect of varying the above parameters on the calculated pressure difference across the model during flow. In particular, the branching ratio was set at known values for dog and human lungs, and diameter and length ratios were independently varied. Known data from dog and human lungs were found to be close to the points predicted by the model where the lines of minimal resistance and minimal entropy production crossed. Other factors which may affect the values of these parameters are discussed.