Pipe Model Theory Research Articles

Dependency of branch diameter growth in young Acer trees on light availability and shoot elongation

Many biomechanical and theoretical studies have been based on the pipe-model theory, according to which a tree is regarded as an assemblage of pipes, each having the same amount of leaf area or leaf mass. However, the physiological mechanisms underlying the theory have not been extensively examined, particularly at the branch level. We analyzed how branches and trunks thickened in nine young Acer mono Maxim. var. marmoratum (Nichols) Hara f. dissectum (Wesmael) Rehder. and A. rufinerve (Siebold & Zucc.) trees. In particular, we examined the roles of light, allocation of photosynthates and shoot heterogeneity. The cross-sectional area (A) of a branch was proportional to cumulative leaf mass or leaf area of the branch, and cumulative cross-sectional area of the daughter branches (SigmaA) above a branching point was equal to the A of the mother branch. These results indicate the validity of the pipe-model theory. However, the theory was invalid for current-year growth of branch cross-sectional area (DeltaA). The DeltaA/SigmaDeltaA for a branching point was greatest (nearly equal to 1) at the crown surface, decreased with crown depth, and tended to increase again at the trunk base, and DeltaA strongly depended on light interception and the yearly increment of leaves on the branch. We examined factors that influenced DeltaA with multiple regression analysis. The ratio of DeltaA of a branch to branch leaf area depended on both relative irradiance and mean current-year shoot length of the branch, suggesting that diameter growth of a branch is determined by the balance between supply of photosynthates, which depends on light interception by the branch, and demand for photosynthates, which is created by the high cambial activity associated with vigorous shoot elongation.

A physical analysis of sap flow dynamics in trees

The aim of this thesis was to analyze the water dynamics of trees by using a dynamic modeling approach. The work is of cross-disciplinary nature: tree water issues which is a subject of whole plant physiology is examined by means of physics. From physical principles five different models were derived. First, to study the sap flow and the water pressure dynamics in the xylem of tree stem, the effect of embolism on the sap flow, and the recovery of embolized conduits; and second, to analyze a sap flow measuring system based on the heat balance method. Sap flow and water pressure dynamics are analyzed with two models that are based on the relation between xylem water tension and changes in the diameter of sapwood. This new approach is advantageous because it offers a way to compare the model predictions with the easily measured diameter changes in intact trees. The model results give new insight into the water dynamics in the stem, stating that pressure propagation is fast, but time lags of a few minutes do exist. These time lags are related to the dimensions and radial elasticity of the stem, and to the sapwood permeability. The results are in agreement with the cohesion-tension theory but partly contradict the pipe model theory of a plant form. The embolism recovery model presents a quantitative analysis of the processes that have been suggested to recover embolized conduits while the water in surrounding conduits is under negative pressure. The model analysis reveals that under normal physiological circumstances, e.g., normal xylem water tension, osmotic pressure of living cells, or diffusion distances, the refilling process is possible if the two sides of the same living cell have different transport properties for solutes. The model for analyzing the performance of a sap flow measuring system reveals that the method, that assumes homogenous temperature field inside the stem, is not appropriate for stems with a diameter larger than a few centimeters. This study has shown that relatively simple models, based on physical principles, are useful in increasing our understanding of the processes and behavior of natural objects. Furthermore, model analysis leading to better understanding of dynamic systems, can guide forthcoming research and enhance the development of new instrumentation.

Construction and maintenance of the optimal photosynthetic systems of the leaf, herbaceous plant and tree: an eco-developmental treatise.

The paper by Monsi and Saeki in 1953 (Japanese Journal of Botany 14: 22-52) was pioneering not only in mathematical modelling of canopy photosynthesis but also in eco-developmental studies of seasonal changes in leaf canopies. Construction and maintenance mechanisms of efficient photosynthetic systems at three different scaling levels--single leaves, herbaceous plants and trees--are reviewed mainly based on the nitrogen optimization theory. First, the nitrogen optimization theory with respect to the canopy and the single leaf is briefly introduced. Secondly, significance of leaf thickness in CO2 diffusion in the leaf and in leaf photosynthesis is discussed. Thirdly, mechanisms of adjustment of photosynthetic properties of the leaf within the herbaceous plant individual throughout its life are discussed. In particular, roles of sugar sensing, redox control and of cytokinin are highlighted. Finally, the development of a tree is considered. Various mechanisms contribute to construction and maintenance of efficient photosynthetic systems. Molecular backgrounds of these ecologically important mechanisms should be clarified. The construction mechanisms of the tree cannot be explained solely by the nitrogen optimization theory. It is proposed that the pipe model theory in its differential form could be a potential tool in future studies in this research area.

A Model for Foliage and Branch Biomass Prediction for Intensively Managed Fast Growing Loblolly Pine

Abstract Prediction models of foliage and branch biomass were derived based on the foliage distribution within the crown and the pipe model theory. Resulting models were fitted to data collected from intensively managed loblolly pine (Pinus taeda L.) plantations in the Lower Coastal Plain and Piedmont of Georgia. We found that diameter outside bark at the base of the live crown, crown height, and crown length are key predictors of foliage biomass. Together they produce reliable predictions of foliage and branch biomass for stands managed under a wide array of silvicultural treatments. The model indicates that an annual fertilization treatment significantly increased foliage and branch biomass in the Lower Coastal Plain. However, in the Piedmont, complete control of competing vegetation significantly increased foliage and branch biomass. A significant fertilization-age interaction for foliage and branch biomass was also detected in Piedmont stands. FOR. SCI. 50(1):65–80.

Interaction between sapwood and foliage area in alpine ash (Eucalyptus delegatensis) trees of different heights.

In native stands of Eucalyptus delegatensis R. T. Baker, sapwood area (As) to foliage area (Af) ratios (As:Af) decreased as tree height increased, contradicting the common interpretation of the Pipe Model Theory as well as the generally observed trend of increasing As:Af ratios with tree height. To clarify this relationship, we estimated sapwood hydraulic conductivity theoretically based on measurements of sapwood vessel diameters and Poiseuille's law for fluid flow through pipes. Despite the observed decrease in As:Af ratios with tree height, leaf specific conductivity increased with total tree height, largely as a result of an increase in the specific conductivity of sapwood. This observation supports the proposition that the stem's ability to supply foliage with water must increase as trees grow taller, to compensate for the increased hydraulic path length. The results presented here highlight the importance of measuring sapwood hydraulic conductivity in analyses of sapwood-foliage interactions, and suggest that measurements of sapwood hydraulic conductivity may help to resolve conflicting observations of how As:Af ratios change as trees grow taller.

Generating 3D sawlogs with a process-based growth model

This paper reports the application of a process-based tree and stand growth model to predictions of stem structure, in particular, the 3D geometry of the stem and its internal knots, in Scots pine ( Pinus sylvestris L.). The model is an extension of an earlier model which predicted tree growth and competition on the basis of carbon balance, and the vertical profile of the stem and crown on the basis of a dynamic interpretation of the pipe model theory. In this study, statistical models are used to create individual branch information from the vertical profile of branch basal area, and the resulting predictions of 3D stem structure are tested against measured trees. The model predictions of the vertical size distribution of external knots in different size classes of trees are realistic, although the model tends to underestimate knot size in the top of the crown especially in dominant trees. As regards the internal knots, the model tends to slightly underestimate the width of the tight knot zone, especially in suppressed trees. The test results suggest that the model is readily applicable to predictions of the effects of stand management on wood quality distribution. Further research topics include incorporating defects other than branch size and quality in the model, as well as testing the model against stand level quality distributions of timber.

The influence of weed competition for light and water on growth and dry matter partitioning of young Pinus radiata, at a dryland site

Above-ground biomass growth and allocation was studied in juvenile Pinus radiata D. Don trees growing with and without competition from the woody weed broom ( Cytisus scoparius L.), at a dryland site, over a 2-year period. The relative significance of broom competition for light and water on seasonal growth of P. radiata was quantified using a growth modelling technique which effectively removed the confounding influences of season and initial tree size from analyses. Allometric regression analysis was used to separate size dependant changes in allocation from functional adjustments made in response to treatment. At the end of the second year, trees growing without broom had an above-ground biomass 25-fold greater than that of trees growing with broom. The presence of broom induced consistently more negative values of predawn needle water potential ( ψ e) in P. radiata over the 2-year period. Treatment differences were most pronounced during the first year when ψ e reached a minimum of −4 MPa in plots with broom, compared to a minimum of −1 MPa in plots without broom. Broom leaf area index increased markedly over the 2-year period, from average values of 0.76 m 2 m −2 in year 1 to 2.26 m 2 m −2 during year 2. During the first year, the strong link between ψ e and fractional biomass growth for trees in plots with broom ( r 2=0.91) suggests growth losses were almost entirely attributable to water deficit. Although ψ e was strongly related to fractional biomass growth over the second year ( r 2=0.71), values across a comparable range of ψ e, were on average 15% lower. Inclusion of a simple shading index explained this difference, suggesting broom restricted tree growth during the second year through competition for both light and water. Compared with P. radiata in weed-free plots, trees growing with broom allocated dry matter preferentially to stems (+8%) at the expense of foliage (−4%), roots (−2%) and branches (−2%). These findings on allocation confirm derivations based on pipe model theory and indicate that the root to shoot ratio in juvenile P. radiata remains relatively stable even under severe water deficit.

Aboveground biomass partitioning and leaf development of Chinese subtropical trees following pruning

To evaluate the effects of pruning on aboveground biomass partitioning of trees and investigate the response of aboveground leaf mass fractions of trees to pruning, a pruning experiment with four pruning intensities (0, 20, 50, and 70%), two pruning seasons (spring and autumn), and four subtropical Chinese tree species was carried out. Pruning treatments were conducted in two successive years. In contrast to the prediction based on the pipe model theory, pruned trees partitioned more of their newly produced aboveground biomass to leaves and less to wood growth in most cases, irrespective of species and pruning seasons. This partitioning pattern was positively correlated with pruning intensity. Pruning reduced the aboveground leaf mass fractions of trees instantaneously. However, due to the increased partitioning of aboveground biomass to leaves following pruning, all pruned trees reached the same aboveground leaf mass fractions as unpruned trees within 1 year after pruning (except for Pinus massoniana after the second pruning). This unexpected biomass partitioning pattern might be attributed to the reuse of open vascular channels in stem which were formerly connected to removed branches and leaves. The increased partitioning of aboveground biomass to leaves following pruning, would benefit pruned trees to alleviate negative pruning effects and recover from the damage.

A Quantitative Analysis on the Root System of a Mangrove, Xylocarpus granatum Koenig

A quantitative analysis was done for the root system of Xylocarpus granatum with reference to the Pipe model theory of tree form proposed by Shinozaki et al. (1964a, b). For this objective, six root systems of X. granatum were excavated using a water-pump, and the roots were weighed for respective diameter categories. A significant allometric relationship between the squared stem-base diameter and the individual root weight was seen. The coefficient of this relationship was statistically regarded to be 1.0, meaning that the individual root weight is proportional to the squared value of stem-base diameter. We also analyzed the relationship between diameter and number of roots for the six sample trees. The number of roots with a given diameter and length was calculated from the weight and the specific gravity of roots. Significant linear relationships were recognized in this relationship for the six sample trees. For all sample trees, the coefficients of relationships were statistically regarded to be - 2.0. From these analyses based on population and individual level, we conclude that the root system of X. granatum obeys the pipe model. Application of this model is helpful in the estimation of root biomass, since individual root weight can be estimated from a proportional constant and the stem-base diameter of a mangrove tree.

Vertical structure of Scots pine crowns in different age and size classes

The pipe model theory postulates a static relationship between foliage mass/area and the cross-sectional area of active pipes in branches and stems. If a regular relationship exists, the theory can be used for modelling growth allocation within crowns, provided that the turnover of active pipes and foliage is understood. The objective of this study was to assess to what extent the assumptions of the pipe model hold true within the crowns of 24 Scots pine sample trees of different age and social position. The results suggest that Scots pine crowns are very regular, but some important modifications to the pipe model assumptions are required. The relative vertical foliage density distribution peaked at about 50% down the live crown regardless of age or social position. The ratio of foliage mass to branch cross-sectional area in the top half of the crown increased from the top downwards. The ratio of cumulative branch cross-sectional area to stem cross-sectional area in the top half seemed to increase with tree vigour or growth rate. The ratio of foliage mass to branch cross-sectional area decreased fast in the lower half of the crown, and this decrease was faster than could be predicted from heartwood formation in the branches. This result may be taken as (1) evidence against the pipe model, or as (2) an indication that active pipes cannot always be identified with sapwood. The latter proposition should be studied further.

Within-tree variation in transpiration in isolated evergreen oak trees: evidence in support of the pipe model theory.

Within-tree variation in sap flow density (SFD) was measured in two isolated evergreen oak (Quercus ilex L.) trees growing in an oak savannah (dehesa) in southwest Spain. Sap flow was estimated by the constant heating method. Three sensors were installed in the trunk of each tree in three orientations: northeast (NE), northwest (NW) and south (S). Sap flow density was monitored continuously from May 18 to September 27, 1993. Daily values of SFD ranged between 500 and 4500 mm3 mm-2 day-1. There were significant differences in SFD between orientations; SFD was higher in the NE and NW orientations than in the S orientation. These differences were noted on both a daily and seasonal time scale, and were less pronounced on cloudy days and at the end of the drought period, when SFD was relatively low. Our results support the idea that branches of trees can be viewed as a collection of small independent plants.

Irregular Heartwood Formation in Maritime Pine (Pinus pinaster Ait):Consequences for Biomechanical and Hydraulic Tree Functioning

As the proportion of sapwood (SW) transformed into heartwood (HW) is irregular both radially and longitudinally in trunks of Maritime pine (Pinus pinaster Ait.), it has been suggested that HW formation is a developmental process, regulated internally within the tree. In trees where stem growth is eccentric due to stem lean or wind action, the number of annual growth rings of SW transformed into HW is greater on the compressed side of the tree. To determine the contribution to bending stiffness, if any, of this prematurely formed HW, four point bending tests were carried out on fresh HW and SW samples taken from the same growth ring, or neighbouring growth rings, at different cross-sectional positions at a height of 2m from six 52-year-old Maritime pines. The mean (±s.e.) modulus of elasticity (E) was 7.6±0.3GPa (longitudinal direction) for all samples. No significant differences in E were found between HW and SW; thus HW does not play a significant mechanical role in bending stiffness. To test a second hypothesis that early HW formation on the compressed side of trees may maintain a constant, optimal volume of SW around the tree, the Pipe Model Theory was applied to 12 52-year-old leaning Maritime pines (angle of lean varied from 0–22°). The surface area (S) of the SW was determined at different heights up the trunk and correlated with crown surface area (Scrown). Regressions between SSWandScrown were highly significant, thereby supporting the theory that HW formation and extension is controlled internally in Maritime pine. HW formation in Maritime pine then serves to maintain an optimal proportion of functional SW which is an important criterion for survival in a species often subjected to severe drought for long periods.

Morphological Analysis of the Rice Root System Based on Root Diameter.

A new method of root-system analysis was designed based on root diameter. Using this method the formation of the root system in the rice cultivar, Hinohikari, was examined and the root system of 40 cultivars at the full heading stage compared. The morphology of the root system was visualized and analyzed based on the pipe model theory. Next, a distribution chart of the root volume was drawn for each the root-diameter classes based on the ideal pipe theory, and the mean root-diameter index which is the root diameter at the distribution peak of the root volume was calculated. In the distribution chart of root volume, the groups of and roots were distinguishable with a boundary at about 0.4 mm. Then, the fine root ratio, which is the of the volume of roots to that of all roots, the thick root index, which is the root diameter at the peak of root-volume distribution in the root group, and the fine root index which is the root diameter at the root-volume distribution peak of the root group, were calculated. The results clearly showed a pattern of increase with the increase in root volume of rice and clarified the atructure of the root system based on root thickness and root volume. The root system was classified into two groups, the root group mainly consisting of crown roots and the root group mainly consisting of lateral roots. Each group had a peculiar distribution peak of root volume, and the two groups were distinguished from each other more clearly after the panicle formation stage. The mean root-diameter index, thick root index, fine root index and fine root ratio were suggested to be good indicators of growth of the root system and of varietal difference in morphology of the root system.

Efficient approximation for building error budgets for process models

Error budgets of process models allow us to partition the uncertainty (estimation error) in model projections caused by propagation of uncertainty in model inputs. Orthogonal polynomials, which are often employed to empirically represent unknown and possibly very complex relationships, have been applied to the development of error budgets by fitting the variance of the projection as a function of the standard errors of the parameter estimates of the process model inputs. Data generation approaches to fit those polynomials have involved some type of factorial design. However, that strategy may become unworkable for process models with many model inputs. In this paper, we propose an efficient method for building error budgets to not only overcome the limitations of factorial arrangements, but also to approximate the results obtainable through those designs. The proposed data generation scheme consists of: (a) repeatedly sampling random, equally-spaced, and equally-probable levels of standard error of the model input parameter estimates to characterize the level of error of their probability distributions; and (b) for each set of standard errors, conducting a number of Monte Carlo runs by sampling random values of input parameter estimates from each of those distributions to obtain projections of the process model and thus estimate the variance of the projection. The variance of the projection is then regressed on the orthogonally-transformed values of the input standard errors. The terms of the orthogonal polynomial are then ranked in decreasing order with respect to their explanatory importance in the function. The characteristics of this sampling scheme are analyzed through intensive supercomputer simulations under different combinations of number of model inputs and sample sizes. An example is conducted for a process forest growth model based on the pipe model theory and the self-thinning rule applied to a stand of red pine ( Pinus resinosa Ait.) growing in the Great Lakes region of North America. The error budget resulting from this example closely resembles the error budget obtained by another study conducted for the same process model and forest stand using a fractional factorial design. It is concluded that the method proposed here provides an efficient strategy for building error budgets for process models with many model inputs.

Modelling of reserve carbohydrate dynamics, regrowth and nodulation in a N2‐fixing tree managed by periodic prunings

ABSTRACTWe used a modified transport resistance approach to model legume tree growth, nodulation and dynamics of reserve carbohydrates after pruning. The model distributes growth between roots and shoots applying the transport resistance approach. Within shoots, growth is divided into leaves, branches and stems applying the pipe model theory. The model also accounts for the metabolic differences of principal N sources, nitrate, ammonium and atmospheric dinitrogen, in a mechanistic way. We compared the simulation results with measured biomass dynamics of Gliricidia sepium (Jacq.) Walp. (Papilionaceae: Robinieae) under humid and subhumid tropical conditions. Comparison showed that the biomass production predicted by the model is close to measured values. Total N2 fixation is also similar to measured values. Qualitatively the model increases the proportion of N2 fixation if roots acquire less mineral N. In the present study, the general form of the model is discussed and compared with similar models. The results encourage the use of this approach for studying biomass dynamics of legume trees under the scheme of periodic prunings. Also, it shows that process‐based models have potential in the simulation of trees disturbed by prunings, herbivory or similar factors.

Needle and stem wood production in Scots pine (Pinus sylvestris) trees of different age, size and competitive status.

We studied effects of tree age, size and competitive status on foliage and stem production of 43 Scots pine (Pinus sylvestris L.) trees in southern Finland. The tree attributes related to competition included foliage density, crown ratio and height/diameter ratio. Needle mass was considered to be the primary cause of growth through photosynthesis. Both stem growth and foliage growth were strongly correlated with foliage mass. Consequently, differences in growth allocation between needles and stem wood in trees of different age, size, or position were small. However, increasing relative height increased the sum of stem growth and foliage growth per unit foliage mass, indicating an effect of available light. Suppressed trees seemed to allocate more growth to stem wood than dominant trees, and their stem growth per unit foliage mass was larger. Similarly, trees in dense stands allocated more growth to stem wood than trees in sparse stands. The results conformed to the pipe model theory but seemed to contradict the priority principle of allocation.

Initializing a model stand for process-based projection.

A stand generation (or initialization) procedure is designed to furnish morphologically plausible model trees for process-based projection. The steps of the initialization are: (i) the generation of the locations of model trees and the tessellation of tree areas; (ii) the sampling of diameters from a target distribution and assignments of those diameters to model-tree locations; (iii) the calculation of the height of each model tree from its assigned diameter and the distances to its neighbors; (iv) the calculation of the crown length of each model tree from its height and distances to its neighbors; and (v) the recalculation of diameter from height and crown length. Components of dry matter are calculated from the model-tree dimensions and pipe-model theory. Process-based projection with the AMORPHYS model is discussed.

Influence of Vertical Foliage Structure on the Distribution of Stem Cross-Sectional Area Increment in Western Hemlock and Balsam Fir

Abstract A set of western hemlock (Tsuga heterophylla Raf. [Sarg.]) and balsam fir (Abies balsamea [L.] Mill.) from western Washington, USA and western Newfoundland, Canada, respectively, were destructively sampled to examine relationships between vertical foliage structure and distribution of stem cross-sectional area increment. A series of stem growth distribution models describing the relationship between amount of foliage area above a given height and stem cross-sectional area increment at that height was derived from the pipe model theory. These models were derived with increasing generality so that two commonly held assumptions could be explicitly tested: (1) linear increase in cross-sectional area increment with increasing foliage area; and (2) constant cross-sectional area increment below base of the live crown. Overall, the models performed very well, accounting for over 80% of the observed variation in cross-sectional area increment for both species. The results show that, while cross-sectional area increment increases with increasing current foliage, the increase is not proportional (i.e., linear). Furthermore, the rate of cross-sectional area increment below base of the live crown was found to increase in the smaller balsam fir trees and decrease in the larger western hemlock trees. These results suggest that cross-sectional area increment per unit foliage below the live crown may change as trees grow. For. Sci. 46(1):86-94.

Analysis of Root Growth Characteristics Based on a Pipe-model Theory and Distribution of Root Nodules in Soybean (Glycine max Merr.).

Based on the pipe-model theory, the root growth characteristics and root nodule distribution in soybean grown in pots were analyzed, using the variety Fukuyutaka and two mutants derived from the variety Enrei, super-nodulating (En6500) and non-nodulating. Fukuyutaka plants were grown under various nitrogen levels, soil hardness, shading levels and bacillus densities, in soil. The ratio of the amount of thin roots which does not conform to the pipe model to that of thick roots which conformed to the model was nearly the same regardless of cultivation conditions. The formation of nodules began in minute radicles of 0.2-0.3 mm diameter class, and about 95% of root nodules were distributed in the thin roots of 1 mm or less. A high correlation was observed between the amount of roots and the number of root nodules in every root-diameter class. However, the big root nodules (larger than 3 mm in diameter) tended to be distributed over all root-diameter classes equally. It was suggested that the distribution of the root nodules with high nitrogen-fixing ability is closely related to the amount of the vascular bundles in each root diameter class.

A process-based growth model for the grass stage pine seedlings

A carbon- and nitrogen-balance model, applying pipe model theory and a modification of functional balance as growth-guiding rules, is presented for the grass stage pine seedlings. Three populations of Pinus merkusii Jungh. et de Vriese, originating from northern and northeastern Thailand, were grown under controlled environment for 47 weeks to obtain parameter information, to evaluate the model performance and to investigate genotypic variation in various characteristics among the populations. Monte Carlo simulations were used to evaluate the sensitivity of the model behaviour to varying parameter values and to calibrate the model for each population. With given sets of parameter values, the simulated biomass development fitted rather well the observed one during the experiment. The two most important parameters determining model performance were within-shoot shading and specific nitrogen uptake rate of fine roots. The fit of simulated versus measured fine roots had a major effect on acceptable model performance in Monte Carlo simulations. Significant variation in biomass growth, nitrogen use efficiency, height, stem diameter, total carbon concentrations of stem and fine roots, and total nitrogen concentrations of needles, transport roots and fine roots was found among the populations. The observed genotypic variation in seedling biomass and stem diameter was consistent with the geographical distribution of the populations while the variation in the rest of the measured characteristics was not. It seems that P. merkusii populations in Thailand are adapted to more site specific conditions rather than climatic conditions alone, and that the variation in biomass growth may result from variation in internal carbon and nitrogen dynamics among the populations.