| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
|
|
|
|||||||
|
|||||||||
Ecology |
UMR Ecologie des Forêts de Guyane, INRA, BP 709, 97379 Kourou, French Guiana; UMR 547 PIAF, INRA, Université Blaise Pascal, 63100 Clermont-Ferrand, France; AgroParisTech, UMR1092 Laboratoire d'étude des Ressources Forêt-Bois (LERFoB), 54000 Nancy, France
Received for publication December 12, 2006. Accepted for publication July 26, 2007.
ABSTRACT
Tree buckling risk (actual height/critical buckling height) is an important biomechanical trait of plant growth strategies, and one that contributes to species coexistence. To estimate the diversity of this trait among wide samples, a method that minimizes damage to the plants is necessary. On the basis of the rarely used, complete version of Greenhill's model (1881, Proceedings of the Cambridge Philosophical Society 4(2): 65–73), we precisely measured all the necessary parameters on a sample of 236 saplings of 16 species. Then, using sensitivity (variance) analysis, regressions between successive models for risk factors and species ranks and the use of these models on samples of self- and nonself-supporting saplings, we tested different degrees of simplification up to the most simple and widely used formula that assumes that the tree is a cylindrical homogeneous pole. The size factor had the greatest effect on buckling risk, followed by the form factor and the modulus of elasticity of the wood. Therefore, estimates of buckling risk must consider not only the wood properties but especially the form factor. Finally, we proposed a simple but accurate method of assessing tree buckling risk that is applicable to a wide range of samples and that requires mostly nondestructive measurements.
Key Words: biomechanics • critical buckling height • French Guiana • risk factor • sapling • stem form • tropical rain forest • trunk volume
Within the scope of forest ecology, plant functional traits must be determined to observe their diversity, to find the existing trade-offs that allow species to coexist (McGill et al., 2006
), and to define species growth strategies. Biomechanical traits of plants are usually studied within different contexts, including studies of the evolution of plant forms (Esser, 1946
; Larson, 1963
; McMahon, 1973
; Niklas, 1988
; Alméras et al., 2004
) and growth (Mattheck, 1990
; Niklas, 1993
; Henry and Aarssen, 1999
; King et al., 2006
). They are also studied at the practical level to develop a better understanding of the mechanical stability of cultivated plants (Brüchert et al., 2000
; Coutand et al., 2000
) or their potential for human use (Beismann et al., 2002
; Kern et al., 2005
). The interactions between mechanical constraints and tree architecture (i.e., the developmental constraint) are increasingly discussed within the context of heterogeneous forest ecology, especially in tropical rainforests characterized by a tremendous diversity of woody plant species (O'Brien et al., 1995
; Sterck and Bongers, 1998
; van Gelder et al., 2006
). In this case, the diversity of tree traits among a wide range of samples of plants and species must be analyzed. Moreover, these traits should be assessed insofar as possible by nondestructive measurements to enable repeated and long-term observations during growth in permanent plots.
One tree trait discussed in many works (Rich et al., 1986
; King, 1987
; Niklas, 1995
; Sterck and Bongers, 1998
; Gavin and Peart, 1999
) is the risk of mechanical buckling under self-weight, usually measured by a safety factor, the ratio between critical buckling height (Hcr) and the actual tree height. The use of such safety factors implies the choice of a particular biophysical constraint, buckling in this case, but other constraints can be analyzed as well, such as uprooting or tree breaks under wind stress (Esser, 1946
; King, 1986
; Spatz and Bruechert, 2000
; Karrenberg et al., 2003
) or under hydraulic stress (Niklas and Spatz, 2004
; Kern et al., 2005
). This implicitly assumes that buckling is ecologically relevant in the studied context. However, using safety factors >1 for self-supporting trees, we highlight the safe situations where the buckling risk is obviously not ecologically significant. Actually, when a safety factor >4, buckling is obviously not a major constraint and plant height is obviously limited by other factors. Therefore, we prefer to use the reciprocal of the safety factor, i.e., the risk factor (RF) that is the ratio between the tree's actual height and its critical buckling height. RF is strictly contained between 0 and 1 for self-supporting plants, and highlights high-risk values, i.e., situations where buckling risk is a major ecological constraint, with a transition from self-supporting to liana habit. Some authors (McMahon, 1973
; Niklas, 1994
) reported very low buckling risks that were fairly constant at the scale of large samples of trees or self-supporting ground plant species. In tropical rainforest understory, the very limited light (Chazdon and Fetcher, 1984
; Montgomery and Chazdon, 2002
) with vertical (and horizontal) light gradients induces tall and slender saplings associated with high buckling risks (Kohyama and Hotta, 1990
; King, 1991
, 1994
). Moreover, a small percentage of nonself-supporting trees are usually observed in this case (see Fig. 1). Finally, based on the diversity of growth patterns, life histories, and architectures found in a tropical rainforest, we expect to find a diversity of buckling risks among species (Bongers and Sterck, 1998
; Sterck and Bongers, 1998
).
|
. This model is based on the beam theory for a tapered, elastic pole, subjected to gravity and with restrained anchorage. Among the many models developed for buckling analysis of tapered beams under complex loadings (Garth Smith, 1988
; Elishakoff and Rollot, 1999
; Li, 2001
), we chose this model, which is very appropriate for tree buckling risk assessment, because the parameters used to specify geometry and loads are adapted to the description of a tree. Greenhill's model considers tree size, taper, mass distribution along the trunk, and wood stiffness. Many authors (King, 1981
; Niklas, 1994
; Claussen and Maycock, 1995
; Sterck and Bongers, 1998
; van Gelder et al., 2006
) have used this model to compute safety factors with more or less implicitly simplified assumptions about trunk taper, mass distribution along the trunk, and wood stiffness, but the relevance of these assumptions has seldom been assessed. Moreover, the definition of RF requires a comparison between the real height and the critical one, where all other factors remain constant. Most authors (Claussen and Maycock, 1995
; Sterck and Bongers, 1998
; van Gelder et al., 2006
) compared trees using one diameter, the same taper, the same mass distribution and, sometimes, the same wood stiffness. When comparing species strategies, the ecological significance of these assumptions is not always obvious. Taper or mass distribution can be easily understood as developmental architectural constraints and should therefore vary among species and with environmental conditions. Wood stiffness variations, closely linked to dry wood density, play a central role in the life-history variation of tree species (van Gelder et al., 2006
). Lastly, the diameter is the usual size parameter in forest science. However, because the issue in this case is to calculate the maximal height the tree could reach with the same investment in support material, the tree should be compared to a pole of the same wood volume or same wood dry mass, rather than the same diameter. This paper compares different methods for estimating buckling risk factors, using Greenhill's model for calculating buckling height. The Hcr and RF will first be computed with the complete model using a sample of 236 saplings from 16 species of the tropical rain forest of French Guiana. The sources of variation of the buckling height will be studied to determine which ones can be disregarded at the intra- and/or interspecific levels. For further applications on large samples on permanent plots where trees cannot be harvested, we will then design proxy variables using nondestructive data for the factors that greatly contribute to Hcr variability. Finally, we will discuss the bias and errors due to the different possible choices. This comparison is based on the consequences of simplifying the assumptions on the Hcr calculation and on the ranking of species according to their RF. The ability of the models to clearly discriminate between saplings known to be self-supporting or not is validated for one species.
MATERIALS AND METHODS
Greenhill's model
The risk for a tree to buckle under its self-weight is calculated by the ratio of its actual height to its critical buckling height Hcr, i.e., the maximal height it could reach with the same volume of material, taking its developmental constraints (mass, tree form, and wood properties) into account. Our reformulation of Greenhill's model leads to Eq. 1:
|
| (1) |
|
| (2) |
|
| (3) |
with defined by Eq. 4:
|
| (4) |
|
| (5) |
|
| (6) |
|
| (7) |
|
| (8) |
|
| (9) |
If the tree is assumed to be a tapered pole of constant density (
) with negligible branch and leaf biomasses, M(z) = V(z), then according to Eq. 6, parameter m equals 2n + 1.
A further simplified version of the model (referred to as the "classical formula" in this work) is often found in the literature (Niklas, 1995
, 1999a
; Sterck and Bongers, 1998
; Falster, 2006
; van Gelder et al., 2006
). In this case, the trunk is considered as a cylinder with, in most cases, a homogeneous distribution of biomass all along the tree so that n = 0 and m = 1 and, thus, c = 1.867. The resolution of Greenhill's model leads to Eq. 10:
|
| (10) |
obtained from our numerical calculations.
In Greenhill's initial model, the parameter used for the diameter is the basal dimension of the stem, but some authors (Sterck and Bongers, 1998
; Sposito and Santos, 2001
; van Gelder et al., 2006
) used the diameter at breast height to define the size of the cylinder used to calculate the Hcr. When applying this formula to our data, we used the diameter measured 1.5 m above the ground (D150), which is easier and more accurate to measure than the basal diameter because of the buttresses, stilt roots, and frequent variations of the stem's circumference near the base of the tree. In Eq. 10, the cylinder defined by the basal diameter may be significantly larger than the one defined by another diameter higher up in the tree, and the Hcr calculated is larger as well.
Plant material and measurements
Sixteen common species of the Guianese tropical rain forest were used (Table 1). The 236 saplings were harvested between 2002 and 2006 at the Paracou Research Station (5°18' N, 52°55' W); see Gourlet-Fleury et al. (2004)
for a complete description of the site. The individuals were chosen to form a representative sample of saplings with D150 ranging from 1 to 7 cm. The mean D150 was 3.9 cm ± 1.9 SD, and the mean height was 7.3 m ± 3.3 SD. The data were collected as follows: after the sapling was cut down, the total length (H) of the main axis was measured. Diameters and weights were measured along the trunk. These data were used to calculate n and m with log–log regressions. To increase the accuracy of the log–log regressions used to calculate n and m, trees were sawed into six parts of equal length, and the two distal parts were again cut into two equal parts. Each of the eight parts was weighed, including trunk, branches, and leaves, and the basal diameter of each part was measured. The determination coefficients for the individual log–log regressions were high (
= 0.952 ± 0.041 for n and 
= 0.973 ± 0.025 for m). All the saplings for which this coefficient was under 0.85 (mostly due to broken saplings) were removed from the analysis. This excluded less than 3% of the sampled saplings. Diameters at the base of each part were also used to calculate trunk volume V, considering each stem segment as a truncated cone. Finally, a 1-cm thick segment of each part was kept and used to measure wood basic density b (oven dry mass/fresh volume). The segments were fully impregnated with water using a vacuum pump, and their volume was measured by the Archimedes principle. The segments were then dried in an oven for 3 days at 103°C and weighed. The basic density is a good proxy of Young's modulus, at least at an interspecific level, as shown for both temperate and tropical trees (Cannell and Morgan, 1987
; van Gelder et al., 2006
). Fournier et al. (2006)
used a compilation of results to quantify the relationship between the modulus of elasticity of green wood (E) and basic density (b):
|
| (11) |
). Some authors (Spatz, 2000
; Spatz and Speck, 2002
) proposed a modified version of Greenhill's model that considers the vertical variation of E. The use of this method requires a numerical computation for each sapling and additional measurements at the individual level to accurately determine the variations in E at the highest parts of the trunk. Because the aim of this work was to determine a method to compute the individual buckling risk for diverse samples, we preferred to use a mean E for each sapling. However, we note that use of a mean E could overestimate Hcr.
|
|
| (12) |
|
| (13) |
|
| (14) |
Finding proxies or mean estimations to avoid destructive measurements
According to the results of the variance analysis for the factors for which the contribution to the total variance is large enough, we tried to find the best relationship between the factor and a combination of H and D150. This was done using the multiple regression tool of the Statistica software (version 7.1, Statsoft France [2006]).
Testing the simplifications of the model
To determine the acceptability of the aforementioned simplifications, we checked their influence on Hcr and RF estimations. The most accurate method, which uses all the factors measured at the individual level (referred to as the "complete model" later), was taken as a reference for comparing Hcr and RF calculated with increasingly simplified models. A good simplified model is characterized by (1) a good correlation with the complete model; (2) no bias, i.e., a slope close to one (the intercept is set to zero); (3) an unchanged ranking of species; and (4) the ability to discriminate between self-supporting and nonself-supporting trees. This last point was then validated with a sample of Tachigali melinonii saplings. These saplings were sampled according to their observed state of mechanical stability (clearly self-supporting or clearly buckled as in Fig. 1) and were included in the sample used in this paper (Table 1). For each validation criterion, we compared the complete model (Eq. 9), our simplified versions, and the classical formula (Eq. 10).
RESULTS
Practical calculation of the constant c (root of a Bessel function)
The constant c, which depends on the allometric parameters through (Eq. 4) can be computed by Eq. 15:
|
| (15) |
for the classically used assumptions of a cylindrical trunk with a homogeneous distribution of biomass, i.e., n = 0 and m = 1, thus = –1/3 and c
= 1.867. The coefficients are shown in Table 2. The small relative errors (the largest one is lower than 2%) confirm the validity of these equations to calculate c.
|
|
Very good relationships were found to predict the volume at a specific level (R2 ranging between 0.956 and 0.999) and also at the interspecific level (R2 = 0.982) (Table 4). For E and L, the intraspecific variance was low so that mean specific values (Table 4) will be tested. Finally, because F has a considerable effect at both the inter- and intraspecific levels, two estimations were tested: a global regression with size parameters (the best relation found was ln F = 1.784 + 0.294 x ln H, R2 = 0.319) and mean specific values.
|
|
|
|
|
Predicting degree of self-support
The complete model and our simplified versions, FmVgr and FmVsr, revealed significant differences between means of RF for both habits, self-supporting and nonself-supporting (Table 7). The mean values given by models FmVgr and FmVsr were close to those given by the complete model. If we used individual values and considered a margin of ±0.1, the RF of some saplings did not correspond (out of the margins) to their habit. Finally, for the classical model, the mean values were higher than for the three other models. The individual values were frequently higher than 1, predicting that almost all the saplings were nonself-supporting. The complete model and our simplified versions revealed an RF significantly lower than 1 for self-supporting saplings and not significantly lower than 1 for nonself-supporting saplings (Fig. 4). The classical model provided higher values with no significant distinction between mean RF of self-supporting saplings and the buckling limit 1.
|
|
This work aims at determining a way to accurately measure the buckling risk factor of saplings. Within the context of forest ecology, the study of plant functional traits and their diversity is a central issue, and the buckling risk has not been accurately studied among sapling populations. We propose a detailed study of this trait assessment and a method to measure it on a wide range of plant populations. The Hcr sensitivity analysis shows the predominance of the size factor V. This result simply expresses the fact that the maximal height that a tree can achieve mainly depends on the amount of material it is made of. It should be noted that the contribution of the volume factor to the total variance of Hcr is directly controlled by the range of sizes of the studied trees. If a wide size range is used, then the volume factor is the main contribution to the variance of Hcr. For instance, Niklas (1994)
studied plants with diameters ranging from 0.003 m to 3 m. The effect of size-independent factors is of much greater biological significance in terms of biomass allocation and optimal mechanical design. The most original result of our work concerns the demonstrated preponderance of the form factor F among size-independent factors. Researchers usually assume that the form factor is constant, i.e., that trees are homogenous and cylindrical (McMahon, 1973
; Claussen and Maycock, 1995
; Niklas, 1995
, 1997
, 1999a
; Sterck and Bongers, 1998
; van Gelder et al., 2006
). As shown in Fig. 2, this assumption does not correspond to the reality of stem form. Moreover, the form factor has large interspecific variability, showing that the distribution of biomass within the tree is an important biomechanical trait of the species. The use of the classical formula (Eq. 10) leads to an underestimation of Hcr and an overestimation of RF, confirmed by the analysis of the subsample of T. melinonii for which the habits are known. Those results are not surprising because the classical formula considers a cylinder, while the majority of the taper values are closer to a cone (n = 1). Obviously, with the same amount of material, a cone can be built higher than a cylinder. Indeed, a cone has both a lower load in its distal part where the lever arm is the biggest and a higher bending inertia in the basal part that is subjected to the highest bending moment. These results are consistent with those of Keller and Niordson (1966)
, who found optimal taper values comprised between 1/3 and 3/2 for unloaded and loaded columns (with an infinitely higher load than its own weight), respectively. Our values of n are closer to 1/3, which corresponds to the sapling situation, i.e., loaded by noninfinite mass. Moreover, when researchers assume cylindrical trees of a given diameter (McMahon, 1973
; Claussen and Maycock, 1995
; Niklas, 1995
, 1997
, 1999a
; Sterck and Bongers, 1998
; van Gelder et al., 2006
), the estimation of Hcr is very sensitive to the choice of the tree diameter (basal, at breast height, etc.), and such choices are rarely discussed. Therefore, disregarding accurate estimations of form factors leads to a bias of the RF estimate because the form factor is both a determining factor of the RF and variable among tree species. Moreover, even if it is not strong, a significant (P < 0.05) relationship has been found (R2 = 0.319) between form factor and the size of saplings. As a result of the small range of sizes in our sample, we were able to use a specific mean value for this factor, but the transposition of this result to a wider sample may not be advisable. Biologists and foresters have been studying stem growth and taper for a long time (Larson, 1963
; Claussen and Maycock, 1995
). Even if some authors (Chiba and Shinozaki, 1994
; Chave et al., 2005
) have reported no change in stem form of saplings over time, there is evidence that this factor is modulated by the immediate environment of the sapling: light, population density, and resource availability (Larson, 1963
; Claussen and Maycock, 1995
; Briand et al., 1999
; Dean et al., 2002
). Fewer data are available on the mass distribution parameter m along the trunk, which integrates biomasses of both the trunk and the branches (wood and leaves). Because trunk wood is quite heavy, we would expect that m could be linked to n. However, such a relationship was not found, and moreover, m is different from 2n + 1, which means that the tree cannot be modeled as a pole of constant density. King and Loucks (1978)
emphasized the importance of mass distribution and developed a model of Hcr based on the ratio R of crown biomass to trunk biomass. Niklas (1994)
underlined and completed the results of King and Loucks (1978)
using an R varying with species and size to compute the Hcr. A correct estimation of the form factor is the main difference between our models and the classical formula, a difference that leads to considerable discrepancies, including a different ranking of species relative to their RF. Therefore, it is essential that any biomechanical study based on buckling analysis acquire data and use existing data about form factors. To avoid the complicated problem of calculating Bessel roots that are not standard mathematical functions, we proposed a simple polynomial fitting of c that will provide practical help for the calculation of F and for further studies.
The load factor was not very sensitive; inter- and intraspecific variabilities are comparable. Therefore, the estimation of L always leads to a slight bias of Hcr and RF calculations. Moreover, interspecific variations can be overlooked without much loss in the accuracy of predictions; the use of a global mean value in the FmVsr model leads to results similar to those given by the complete model. However, the stable value of L should depend on the studied situation. Finally, we chose to use a mean specific value for E. Although less sensitive than the form factor, the wood modulus of elasticity is involved in the variability of biomechanical stability, as emphasized by van Gelder et al. (2006)
. We found greater inter- than intraspecific differences for E. This is consistent with other works (Wiemann and Williamson, 1988
; Barbosa and Fearnside, 2004
; Muller-Landau, 2004
). Wood density is known to depend on the ecology of the species (Wiemann and Williamson, 1989
; Suzuki, 1999
; Woodcock and Shier, 2003
; Muller-Landau, 2004
), with less dense and stiff wood on pioneer, fast-growing species. Thus, the use of a mean specific value seems acceptable and requires only a few destructive measurements because of the low intraspecific variability. However, we stress that the actual measured factor is wood basic density and not Young's modulus of elasticity. We used a relationship between wood modulus of elasticity along the grain and basic density, as is typical in cellular materials and wood science (Kollmann and Cote, 1968
), and made sure that it was very good at predicting interspecific variations of wood stiffness (Fournier et al., 2006
). Wood basic density is linked to Young's modulus, but this relationship is subject to exceptions (Guitard, 1987
) because of the ultrastructure of wood cells (the microfibril angle may differ among woods of similar basic density, resulting in differences in mechanical properties as well). Wood basic density is less variable than Young's modulus, and we may thus underestimate the participation of this factor in the total variance of Hcr. Nevertheless, each time we had the opportunity to directly verify the accuracy of the estimation for tropical green wood, the predicted value of E was very close to the measured value (Clair et al., 2003
). However, because many studies reveal variations in the modulus of elasticity with ontogeny (Rueda and Williamson, 1992
; de Castro et al., 1993
; Woodcock and Shier, 2003
), environment (Fearnside, 1997
; Suzuki, 1999
; Baker et al., 2004
), or the ecology of the species (Wiemann and Williamson, 1989
; Muller-Landau, 2004
), it is advisable to make new measurements for each new population studied. Using a mean interspecific value of E in the FmVsr model does not reduce the precision of calculations, with the exception, once again, of the RF calculation. It is therefore acceptable to use a mean interspecific value to avoid destructive measurements. Some authors have also reported that wood properties change from pith to bark (Wiemann and Williamson, 1989
; Woodcock and Shier, 2002
). Changes may also occur because of reaction wood production. We measured wood properties on segments representing the whole stem section, thus giving us a "global modulus of elasticity." The mechanically correct measurement of the equivalent modulus of elasticity would have required us to consider each different layer and its relative contribution to the flexural inertia, but because E is not the main contributing factor to Hcr and because of the high interspecific variations, we could use this method without inducing too large of an error.
Within the framework of this study, the analysis of T. melinonii saplings clearly shows that trees in a forest are not always self-supporting. Thus, RF values larger than 1 are not only due to an artifact, as suggested by Niklas (1994)
, but they can reveal a nonself-supporting habit as part of a growth strategy. Assumptions made in our simplified models do not considerably change the ranking of species according to their biomechanical strategy. This is not the case with the classical formula. However, the accuracy of the estimation by Greenhill's model was not obvious because there are many underlying assumptions: consistently circular cross sections, branch weights assumed to act similarly to the trunk with no additional bending due to asymmetric development, perfectly rigid anchorage, wood variability, etc. We verified that Greenhill's model by itself is a good estimation of the self-supporting habit. When comparing self-supporting and nonself-supporting trees, we found that the model accurately discriminated between the different habits. This type of discussion about the performance of buckling mechanical models rarely occurs in the literature (Tateno and Bae, 1990
). We finally proposed a better way to estimate Hcr and RF; in contrast to the widely used classical formula, our method emphasizes the importance of form factor values and variability. Concerning biomechanical ecological studies of tree species, we suggest that the height and diameter of each individual be measured nondestructively, then the mean values for E, L, and F and the relationships between V and H and D150 should be estimated using smaller samples of harvested trees for each population. In any case, the use of a global relationship for V does not induce a large bias. Moreover, the method developed (i.e., the analysis of variance among the samples that justifies the choice of estimations for each factor and allows a classification of factors according to their sensitivity) is easy to reproduce in other situations, for example, in comparisons of different plant forms in phylogenetic studies. Further studies will focus on using this method to monitor and analyze the biomechanical diversity of tree species in permanent plots and to understand the relationship between biomechanical traits and species ecology.
FOOTNOTES
1 The authors would like to thank everyone who contributed to data collection: M.-F. Prévost and G. Elfort (IRD) and P. Petronelli (CIRAD-forêt) for species identification; I. Godard, D. Jullien, C. Moulia, B. Moulia, E. Nicolini, J. Ruelle, A. Thibaut, and all students who took part in the ENGREF (Kourou) "Tropical rainforests" training courses, including P. Blanquet, G. Crouzet, M. Derycke, P. Fourreau, M. Ghestem, G. Jardinier, M. Jeannesson, C. Levesque, C. Martin, E. Petit, P. Semanaz, B. Soukupova, and B. Zubieta; G. Wagman for improving the language; and CIRAD-forêt for the opportunity to work on its long-term experiment in Paracou. This work was financially supported by the French ANR project, Woodiversity. They are also grateful to the anonymous reviewers for constructive, helpful comments. 
5 Author for correspondence (jaouen_g{at}kourou.cirad.fr
)
LITERATURE CITED
Alméras T. Costes E. Salles J.-C.. 2004. Identification of biomechanical factors involved in stem shape variability between apricot tree varieties. Annals of Botany 93: 455-468..
Baker T. R. Phillips O. L. Malhi R. Almeida S. Arroyo L. Di Fiore A. Erwin T. Killeen T. J. Laurance S. G. Laurance W. F. Lewis S. L. Lloyd J. Monteagudo A. Neill D. A. Patino S. Pitman N. C. A. Silva J. N. M. Vasquez Martinez R.. 2004. Variation in wood density determines spatial patterns in Amazonian forest biomass. Global Change Biology 10: 545-562..[CrossRef][ISI]
Barbosa R. I. Fearnside P. M.. 2004. Wood density of trees in open savannas of the Brazilian Amazon. Forest Ecology and Management 199: 115-123..[CrossRef][ISI]
Beismann H. Schweingruber F. Speck T. Körner C.. 2002. Mechanical properties of spruce and beech wood grown in elevated CO2. Trees—Structure and Function 16: 511-518..
Bongers F. Sterck F. J.. 1998. Architecture and development of rainforest trees: responses to light variation. In D. M. Newbery, [ed.], Dynamics of tropical communities, 125-162. Blackwell Science, Oxford, UK..
Briand C. H. Campion S. M. Dzambo D. A. Wilson K. A.. 1999. Biomechanical properties of the trunk of the devil's walking stick (Aralia spinosa, Araliaceae) during the crown-building phase: implications for tree architecture. American Journal of Botany 86: 1677-1682..
Brüchert F. Becker G. Speck T.. 2000. The mechanics of Norway spruce [Picea abies (L.) Karst]: mechanical properties of standing trees from different thinning regimes. Forest Ecology and Management 135: 45-62..[CrossRef][ISI]
Cannell M. G. R. Morgan J.. 1987. Young's modulus of sections of living branches and tree trunks. Tree Physiology 3: 355-364..[ISI][Medline]
Chave J. Andalo C. Brown S. Cairns M. A. Chambers J. Q. Eamus D. Fölster H. Fromard F. Higuchi N. Kira T. Lescure J.-P. Nelson W. Ogawa H. Puig H. Riéra B. Yamakura T.. 2005. Tree allometry and improved estimation of carbon stocks and balance in tropical forests. Oecologia 145: 87-99..[CrossRef][ISI][Medline]
Chazdon R. L. Fetcher N.. 1984. Light environments of tropical forests. In E. Medina, H. A. Mooney, C. Vazquez-Yanes, [eds.], Physiological ecology of plants of the wet tropics, 27-36. Junk, The Hague, Netherlands..
Chiba Y. Shinozaki K.. 1994. A simple mathematical model of growth pattern in tree stems. Annals of Botany 73: 91-98..
Clair B. Fournier M. Prévost M.-F. Beauchêne J. Bardet S.. 2003. Biomechanics of buttressed trees: bending strains and stresses. American Journal of Botany 90: 1349-1356..
Claussen J. W. Maycock C. R.. 1995. Stem allometry in a North Queensland tropical rainforest. Biotropica 27: 421-426..[CrossRef][ISI]
Coutand C. Julien J.-L. Moulia B. Mauget J. C. Guitard D.. 2000. Biomechanical study of the effect of a controlled bending on tomato stem elongation: global mechanical analysis. Journal of Experimental Botany 51: 1813-1824..
de Castro F. Williamson G. B. de Jesus R. M.. 1993. Radial variation in the wood specific gravity of Joannesia princeps: the roles of age and diameter. Biotropica 25: 176-182..[CrossRef][ISI]
Dean T. J. Roberts S. D. Gilmore D. W. Maguire D. A. Long J. N. O'Hara K. L. Seymour R. S.. 2002. An evaluation of the uniform stress hypothesis based on stem geometry in selected North American conifers. Trees—Structure and Function 16: 559-568..
Elishakoff I. Rollot O.. 1999. New closed-form solutions for buckling of a variable stiffness column by Mathematica. Journal of Sound and Vibration 224: 172-182..[CrossRef][ISI]
Esser M. H. M.. 1946. Tree trunks and branches as optimum mechanical supports of the crown. I. The trunk. Bulletin of Mathematical Biophysics 8: 65-74..[CrossRef]
Falster D. S.. 2006. Sapling strength and safety: the importance of wood density in tropical forests. New Phytologist 171: 237-239..[CrossRef][ISI][Medline]
Fearnside P. M.. 1997. Wood density for estimating forest biomass in Brazilian Amazonia. Forest Ecology and Management 90: 59-87..[CrossRef][ISI]
Fournier M. Stokes A. Coutand C. Fourcaud T. Moulia B.. 2006. Tree biomechanics and growth strategies in the context of forest functional ecology. In A. Herrel, T. Speck, N. P. Rowe, [eds.], Ecology and biomechanics. A mechanical approach to the ecology of animals and plants. Taylor and Francis, CRC Press, Boca Raton, Florida, USA..
Garth Smith W. G.. 1988. Analytic solutions for tapered columns buckling. Computers and Structures 28: 677-681..[CrossRef]
Gavin D. G. Peart D. R.. 1999. Vegetative life history of a dominant rain forest canopy tree. Biotropica 31: 288-294..[CrossRef][ISI]
Gourlet-Fleury S. Guehl J.-M. Laroussinie O.. 2004. Ecology and management of a neotropical rainforest. Lessons drawn from Paracou, a long-term experimental research site in French Guiana. Elsevier, Paris, France..
Greenhill A. G.. 1881. Determination of the greatest height consistent with stability that a vertical pole or mast can be made, and the greatest height to which a tree of given proportions can grow. Proceedings of the Cambridge Philosophical Society 4: 65-73..
Guitard D.. 1987. Mécanique du matériau bois et composites. Cépaduès, Toulouse, France..
Henry H. A. L. Aarssen L. W.. 1999. The interpretation of stem diameter–height allometry in trees: biomechanical constraints, neighbour effects, or biased regressions?. Ecology Letters 2: 89-97..[CrossRef][ISI]
Karrenberg S. Blaser S. Kollmann J. Speck T. Edwards P. J.. 2003. Root anchorage of saplings and cuttings of woody pioneer species in a riparian environment. Functional Ecology 17: 170-177..[CrossRef]
Keller J. B. Niordson F. I.. 1966. The tallest column. Indiana University Mathematics Journal (ex. Journal of Mathematics and Mechanics) 16: 433-446..[ISI]
Kern K. A. Ewers F. W. Telewski F. W. Koehler L.. 2005. Mechanical perturbation affects conductivity, mechanical properties and aboveground biomass of hybrid poplars. Tree Physiology 25: 1243-1251..[ISI][Medline]
King D. A.. 1981. Tree dimensions: maximizing the rate of height growth in dense stands. Oecologia 51: 351-356..[CrossRef][ISI]
King D. A.. 1986. Tree form, height growth, and susceptibility to wind damage in Acer saccharum. Ecology 67: 980-990..[CrossRef][ISI]
King D. A.. 1987. Load bearing capacity of understory treelets of a tropical wet forest. Bulletin of the Torrey Botanical Club 114: 419-428..[CrossRef][ISI]
King D. A.. 1991. Correlations between biomass allocation, relative growth rate and light environment in tropical forest saplings. Functional Ecology 5: 485-492..[CrossRef]
King D. A.. 1994. Influence of light level on the growth and morphology of saplings in a Panamanian forest. American Journal of Botany 81: 948-957..[CrossRef][ISI]
King D. A. Davies S. J. Tan S. Noor N. S. M.. 2006. The role of wood density and stem support costs in the growth and mortality of tropical trees. Journal of Ecology 94: 670-680..[CrossRef][ISI]
King D. A. Loucks O. L.. 1978. The theory of tree bole and branch form. Radiation and Environmental Biophysics 15: 141-165..[CrossRef][ISI][Medline]
Kohyama T. Hotta M.. 1990. Significance of allometry in tropical saplings. Functional Ecology 4: 515-521..[CrossRef]
Kollmann F. F. P. Cote W. A. J.. 1968. Principles of wood science and technology. I. Solid wood. Allen & Unwin, Berlin, Germany..
Larson P. R.. 1963. Stem form development of forest trees. Forest Science Monograph 5: 1-42..
Li Q. S.. 2001. Exact solutions for buckling of non-uniform columns under axial concentrated and distributed loading. European Journal of Mechanics, A/Solids 20: 485-500..[CrossRef]
Mattheck C.. 1990. Why they grow, how they grow: the mechanics of trees. Arboricultural Journal 14: 1-17..
McGill B. J. Enquist B. J. Weiher E. Westoby M.. 2006. Rebuilding community ecology from functional traits. Trends in Ecology and Evolution 21: 178-185..[CrossRef]
McMahon T. A.. 1973. Size and shape in biology. Elastic criteria impose limits on biological proportions, and consequently on metabolic rates. Science 179: 1201-1204..
Montgomery R. A. Chazdon R. L.. 2002. Light gradient partitioning by tropical tree seedlings in the absence of canopy gaps. Oecologia 131: 165-174..[CrossRef][ISI]
Muller-Landau H. C.. 2004. Interspecific and inter-site variation in wood specific gravity of tropical trees. Biotropica 36: 20-32..[ISI]
Niklas K. J.. 1988. Biophysical limitations on plant form and evolution. In L. D. Gottlieb, S. K. Jain, [eds.], Plant evolutionary biology, 185-220. Chapman and Hall, London, UK..
Niklas K. J.. 1993. Influence of tissue density-specific mechanical properties on the scaling of plant height. Annals of Botany 72: 173-179..
Niklas K. J.. 1994. Interspecific allometries of critical buckling height and actual plant height. American Journal of Botany 81: 1275-1279..[CrossRef][ISI]
Niklas K. J.. 1995. Plant height and the properties of some herbaceous stems. Annals of Botany 75: 133-142..
Niklas K. J.. 1997. Mechanical properties of black locust (Robinia pseudoacacia L.) wood. Size and age-dependent variations in sap- and heartwood. Annals of Botany 79: 265-272..
Niklas K. J.. 1999a. Changes in the factor of safety within the superstructure of a dicot tree. American Journal of Botany 86: 688-696..
Niklas K. J.. 1999b. The mechanical role of bark. American Journal of Botany 86: 465-469..
Niklas K. J. Spatz H.-C.. 2004. Growth and hydraulic (not mechanical) constraints govern the scaling of tree height and mass. Proceedings of the National Academy of Sciences, USA 101: 15661-15663..
O'Brien S. T. Hubbell S. P. Spiro P. Condit R. Foster R. B.. 1995. Diameter, height, crown, and age relationship in eight neotropical tree species. Ecology 76: 1926-1939..[CrossRef][ISI]
Rich P. M. Helenurm K. Kearns D. Morse S. R. Palmer M. W. Short L.. 1986. Height and stem diameter relationships for dicotyledonous trees and arborescent palms of Costa Rican tropical wet forest. Bulletin of the Torrey Botanical Club 113: 241-246..[CrossRef][ISI]
Rueda R. Williamson G. B.. 1992. Radial and vertical wood specific gravity in Ochroma pyramidale (Cav. ex Lam.) Urb. (Bombacaceae). Biotropica 24: 512-518..[CrossRef][ISI]
Spatz H.-C.. 2000. Greenhill's formula for the critical Euler buckling length revisited. Proceedings of the 3rd Plant Biomechanics Conference, Freiburg-Badenweiler, Georg Thieme Verlag Stuttgart, H.-C. Spatz and T. Speck..
Spatz H.-C. Bruechert F.. 2000. Basic biomechanics of self-supporting plants: wind loads and gravitational loads on a Norway spruce tree. Forest Ecology and Management 135: 33-44..[CrossRef][ISI]
Spatz H.-C. Speck O.. 2002. Oscillation frequencies of tapered plant stems. American Journal of Botany 89: 1-11..
Sposito T. C. Santos F. A. M.. 2001. Scaling of stem and crown in eight Cecropia (Cecropiaceae) species of Brazil. American Journal of Botany 88: 939-949..
Sterck F. J. Bongers F.. 1998. Ontogenetic changes in size, allometry, and mechanical design of tropical rain forest trees. American Journal of Botany 85: 266-272..[Abstract]
Suzuki E.. 1999. Diversity in specific gravity and water content of wood among Bornean tropical rainforest trees. Ecological Resear
20 range. c
