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Keywords:

  • hydraulic limitation theory;
  • leaf-specific conductance;
  • pipe-model;
  • plant hydraulic architecture;
  • WBE

Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. Can vessel tapering compensate for height growth?
  5. Are the predictions of WBE confirmed by observations?
  6. Does acclimation to increased hydraulic resistance affect forest growth?
  7. Concluding remarks
  8. Acknowledgements
  9. References
  10. Appendix
  • 1
    The hydraulic limitation theory proposes that the decline of forest productivity with age is a consequence of the loss of whole-plant and leaf-specific hydraulic conductance with tree height caused by increased friction. Recent theoretical analyses have suggested that tapering (the broadening of xylem vessel diameter from terminal branches to the base of the stem) could compensate completely for the effect of tree height on hydraulic conductance, and thus on tree growth.
  • 2
    The data available for testing this hypothesis are limited, but they do not support the implication that whole-tree and leaf-specific hydraulic conductance are generally independent of tree height. Tapering cannot exclude hydraulic limitation as the principle mechanism for the observed decline in growth.
  • 3
    Reduction of the leaf-to-sapwood area ratio, decreased leaf water potential, loss of leaf-cell turgor, or osmotic adjustments in taller trees could reduce the effect of increased plant hydraulic resistance on stomatal conductance with height. However, these mechanisms operate with diminishing returns, as they infer increased costs to the tree that will ultimately limit tree growth. To understand the decline in forest growth, the effects of these acclimation mechanisms on carbon uptake and allocation should be considered.

Introduction

  1. Top of page
  2. Summary
  3. Introduction
  4. Can vessel tapering compensate for height growth?
  5. Are the predictions of WBE confirmed by observations?
  6. Does acclimation to increased hydraulic resistance affect forest growth?
  7. Concluding remarks
  8. Acknowledgements
  9. References
  10. Appendix

The growth rate of trees declines with age. The leading hypothesis to explain this is decreased hydraulic conductance as a consequence of increased tree height (Ryan & Yoder 1997). Assuming a simple pipe model of the plant's hydraulic system (a leaf attached to a unit stem and a unit root pipe: Shinozaki et al. 1964b; Shinozaki et al. 1964a; Jarvis 1975), the hydraulic path length – and along with it, the resistance of the plant's hydraulic system caused by friction – increases with tree height (Whitehead & Hinckley 1991). Increased stomatal closure to prevent a breakdown of the hydraulic transport system because of cavitation (Ryan & Yoder 1997), or lower leaf water potentials triggering leaf-cell turgor loss or osmotic adjustment, would reduce daily net photosynthesis. Less growth would be the ultimate result, as the other carbon costs (respiration, foliage production and root growth) vary little, or increase slightly with stand age (Ryan & Waring 1992; Mencuccini & Grace 1996b). Increased respiration by itself, as a consequence of the increasing amount of living woody biomass with tree size (Yoda et al. 1965), cannot explain the decline in forest productivity. Maintenance respiration of the sapwood accounts for only 5–12% of the annual C fixation (Ryan et al. 1995), and the larger fraction of respiration is associated with growth processes that generally decline as growth declines (Ryan & Waring 1992).

The hydraulic limitation theory has been controversial (Becker, Meinzer & Wullschleger 2000b; Bond & Ryan 2000; Mencuccini & Magnani 2000). From the formal hydraulic model of Whitehead, Edwards & Jarvis (1984), one might expect acclimation of the ratio between supporting sapwood area and transpiring leaf area. Such an acclimation could result in homeostatic balance between the plant's transport capacity and transpirational demand of the leaves, to either minimize or compensate for the potential hydraulic limitation of stomatal conductance caused by height. Tree height would still affect tree growth, because hydraulic adjustment would increase the relative share of C production that is allocated to sapwood construction and maintenance, thereby diminishing the returns per unit C investment.

Recent theoretical analyses (West, Brown & Enquist 1999; Enquist 2002) have suggested that the observed tapering of xylem conduits, i.e. the broadening of xylem vessels basipetally (Zimmermann 1978; Pothier, Margolis & Waring 1989), could produce hydraulic homeostasis of the plant. This conclusion is derived from a model of plant branching networks based on the assumption of minimized internal transport resistance while maximizing the plant's surface area. This model describes the plant's architecture and hydraulic system as a self-similar fractal, and in a more detailed way than the pipe model, as it allows for branching and for changes in the xylem anatomy between different branch segments. A basipetal increase in conduit diameter would permit the total resistance of a tube running from trunk to petiole to remain constant, independent of tube length (West et al. 1999).

Here, the published evidence relevant to the scaling of hydraulic resistance, as predicted by the model of West et al. (1999), is reviewed. I discuss whether the data support the assumptions of the fractal model and the conclusions drawn from it. The aim is to evaluate whether tapering would suffice to discard the hydraulic limitation theory as the most likely explanation for the observed age-related decline in forest productivity.

Can vessel tapering compensate for height growth?

  1. Top of page
  2. Summary
  3. Introduction
  4. Can vessel tapering compensate for height growth?
  5. Are the predictions of WBE confirmed by observations?
  6. Does acclimation to increased hydraulic resistance affect forest growth?
  7. Concluding remarks
  8. Acknowledgements
  9. References
  10. Appendix

The architecture of the branching network in the general plant model of West, Brown & Enquist (1999) (hereafter WBE) is described by the scaling of branch diameter, branch length, and conduit diameter of adjacent branch segments of the tree according to simple power laws (Appendix, equation A1). Three parameters define the relationships between branch diameter, branch length, and conduit diameter of adjacent branch levels. Conduits are assumed to operate as if they were tightly bundled, not interconnected, and to have a uniform diameter within a branch segment. They form a continuous tube, represented by a linear series of vessels, conserving the total numbers of tubes in each segment. Thus total plant hydraulic resistance depends linearly on the total resistance of a single tube. Total plant resistance can be approximated by the Hagen–Poiseuille formula, which describes capillary flow (Nobel 1983; Becker, Gribben & Lim 2000a). Tapering is included in the model by scaling conduit diameter such that it increases basipetally, thus allowing the ratio of conductive to non-conductive tissue to vary between branch segments.

An intuitive way to see why tapering can compensate for the longer path length of the tree with increasing tree height in the WBE model is to look at total path length L and resistance Z with increasing number of segments. In the WBE model length and conduit diameter, and thus resistance, of the petiole are assumed to be invariant, thus the total path length and resistance of a single tube can be described as simple sums of petiole length lN and resistance ZN, respectively, with N the number of branching levels in the network:

  • image((eqn 1) )

and

  • image((eqn 2) )

The scaling factors m and i linking the two adjacent levels k and k – 1 can be derived from the basic equations of WBE (see Appendix for derivations):

  • image((eqn 3) )
  • image((eqn 4) )

where n is the number of daughter branches per branch, and inline image and β are the scaling factors for segment conduit diameter and length, respectively. For simplicity, n, inline image and β are assumed to be level-independent, that is, constant for all branch segments (although, as shown below, this is not a requirement). Three conclusions can be drawn from the limiting behaviour of length and resistance for a large number of branching levels.

  • 1
    For a network without tapering (inline image = 0, pipe model), hydraulic resistance grows in proportion to path length (m = i), consistent with the hydraulic limitation theory in its original form (Yoder et al. 1994, Ryan, Binkley & Fownes 1997).
  • 2
    Any (level-independent) tapering (inline image > 0) will cause hydraulic resistance to grow more slowly than path length, as then m always exceeds i. The extent to which this influences the capacity of trees to compensate for height growth is sensitive to assumptions about the scaling of tapering and branch length (Fig. 1; Becker et al. 2000a).
  • 3
    Total plant hydraulic resistance (equation 2) approaches an asymptotic value with increasing number of segments when i < 1. Unlike hydraulic resistance, path length (equation 1) always increases in the WBE model, because m ≥ 1 for any plausible value of β. Thus total plant hydraulic resistance becomes independent of total path length when the tapering exponent inline image fulfils the inequality inline image > β/6 (equation 4). Two assumptions are introduced in the WBE model: that biomechanical constraints and tapering are uniform; and that the branching network is volume-filling, leading to the special case of a self-similar fractal model with β = 1 (West et al. 1999). Equation 4 thus implies for the WBE model that inline image would have to be greater than or (in the limiting case of very large trees) equal to 1/6.
image

Figure 1. Total plant hydraulic resistance (Z, in m−1 s−1 × 10−15) as a function of total path length (L, in m), assuming that lN = 0·05 m, aN = 10 µm, and β = 1 for different tapering coefficients inline image. 0 is the pipe-model theory; 0·167 WBE, 0·09 and 0·29 are estimates of the 95% confidence interval of the tapering parameter estimated by Becker et al. (2001), assuming β = 1.

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The original version of WBE (West, Brown & Enquist 1997) minimized the energy required to distribute resources without any variation in xylem diameter. Tapering was introduced in a subsequent version of the model, ostensibly because hydraulic constrictions would otherwise preclude the existence of tall trees, and cause leaves high in the canopy to receive less water than leaves deeper in the canopy (West et al. 1999). However, WBE describes a fractal-like structure, so that all leaves have equal supply irrespective of their position within the canopy and even with zero tapering, as all tubes have equal path length.

Evidence for even quite small trees (Zimmermann 1978; F. Magnani, unpublished data; cf. Mencuccini 2002) indicates that the degree of tapering diminishes basipetally. Also, James et al. (2003) found that conduits tapered within the crown of several tropical tree species, but lumen diameters were virtually constant below the base of the crown. These results raise concerns that the assumption of a uniform tapering might be too strong (Becker & Gribben 2001; Mencuccini 2002). A declining rate of tapering does not necessarily invalidate tapering as a mechanism to avoid the adverse effects of tree height, contrary to the assumption of Becker & Gribben (2001). In the derivation of equation 4, no assumptions were made that require the coefficients to be level-independent (see Appendix). In fact, any combination of n, β and inline image that results in i < 1 for a particular segment k will achieve this. It is not necessary to assume fractality to minimize the hydraulic resistance and to decouple total plant hydraulic resistance from plant height, agreeing with the more mathematically exact account of Dodds, Rothman & Weitz (2001).

Recent results from McCulloh et al. (2003) suggest for vines and ring-porous trees that the total number of conduits per segment varies between adjacent segments, conflicting with the assumption made in WBE. This would be expected from the application of Murray's law to the hydraulic architecture of plants. If conduits serve transport purposes only (as in ring porous trees), the construction cost of conduits can be assumed to be proportional to conduit volume (McCulloh et al. 2003). Murray's law applied to plant vascular systems suggests that, for such a case, the original pipe model (inline image = 0 or no taper) would be optimal for a constant number of conduits. Any degree of taper without variation in total tube number would be less efficient, and would therefore violate the energy minimization principle. If the number of conduits were allowed to vary between segments in the WBE model, as suggested by the data of McCulloh et al. (2003), the required taper to decouple hydraulic resistance from path length cannot be calculated by equations 1–4. Whole-plant hydraulic conductance then becomes difficult to predict, limiting the capacity of WBE to predict the whole-plant effect of tapering on hydraulic conductance.

Are the predictions of WBE confirmed by observations?

  1. Top of page
  2. Summary
  3. Introduction
  4. Can vessel tapering compensate for height growth?
  5. Are the predictions of WBE confirmed by observations?
  6. Does acclimation to increased hydraulic resistance affect forest growth?
  7. Concluding remarks
  8. Acknowledgements
  9. References
  10. Appendix

To judge whether WBE adequately describes the hydraulic architecture of trees, the distribution of hydraulic resistance within the tree needs to be modelled correctly, and the model predictions of the scaling between size parameters (e.g. diameter) and hydraulic resistance should agree with relevant measurements. Unfortunately, although plant hydraulic architecture has long been identified as an important subject (Zimmermann 1978), surprisingly few relevant data are available. Only a few studies have been designed to exclude the effect of differences in environmental factors (e.g. understorey vs canopy trees, unequal access to groundwater resources; Bond & Ryan 2000), and few analyses allow the assessment of whole-plant hydraulic resistance (Mencuccini 2002).

West et al. (1999) claimed that the degree of tapering predicted by their model agrees with the observed distribution of hydraulic resistance in the two Acer species analysed by Yang & Tyree (1993). However, no data on the scaling of conduit tapering and branch length are available from Yang & Tyree (1993) to confirm that the observed agreement is a consequence of correct assumptions of the WBE model. In a re-analysis of the data of Zimmermann (1978), Becker et al. (2001) found that the parameter estimate for the tapering parameter inline image is sensitive to the assumptions made on the (unknown) scaling of segment length. As only a slight deviation will yield strongly different results in terms of overall hydraulic resistance (Fig. 1), these data cannot indicate if the predicted degree of tapering is sufficient to compensate for height growth.

Becker et al. (2001) showed that WBE is potentially useful in interpreting conduit tapering by producing a good fit of the model to the data of Zimmermann (1978), although the estimated value of the tapering parameter inline image was uncertain. A meta-analysis of existing studies on tree hydraulic architecture by Mencuccini (2002) corroborates the capacity of WBE to predict the scaling of hydraulic conductivity of segments to the segment's diameter, illustrating the fact that larger stems are more permeable (Mencuccini & Grace 1996a; Becker et al. 2000b). However, the data do not support the scaling of whole-tree hydraulic resistance to diameter (and total plant biomass): the observed scaling coefficients are smaller than predicted (Table 1), and whole-plant hydraulic conductance declines with increasing size, contradicting the WBE model.

Table 1.  Allometric scaling of hydraulic conductivity of stem-segment hydraulic conductance to stem-segment diameter, and above-ground hydraulic conductance to diameter at breast height and above-ground biomass based on reduced-major-axis regression (cf. Mencuccini 2002)
SpeciesSegment diameterStem diameterBiomassAuthors
bSEPbSEPbSEP
  1. Analyses using ordinary least-squares regression obtained similar results.

  2. Abbreviations: b = regression coefficient; SE = standard error; P = significance of t-test for difference between regression coefficient and theoretical slopes predicted by the WBE model (2·67 for segment diameter; 2·0 for stem diameter; 0·75 for biomass).

  3. ns = Not significantly different, P > 0·05; *P < 0·05; **P < 0·01; ***P < 0·001.

Acer rubrum2·660·05ns1·720·08***0·650·03**Yang & Tyree (1994)
Acer saccharum   1·680·12*0·650·05nsTyree & Sperry (1988), Yang & Tyree (1994)
Pinus sylvestris2·630·08ns1·230·14***0·500·05***Mencuccini & Grace (1996a)
Pinus banksiana   1·340·22*0·520·09*Pothier et al. (1989)
All four species   1·320·06***0·540·02*** 

Smaller scaling coefficients than predicted by WBE do not necessarily suggest that the water supply to leaves (leaf-specific hydraulic conductance) is reduced with height, as data also support the hypothesis that leaf-to-sapwood area ratio declines with tree height (Whitehead et al. 1984; Mencuccini & Magnani 2000,McDowell et al. 2002). Yet there is a significant reduction of leaf-specific hydraulic conductance with height in three out of the four studies summarized in Table 1; in the fourth, the sample size was too limited to draw any significant conclusion (Mencuccini et al. 2000). Similar results were obtained for Pinus ponderosa (Ryan et al. 2000) and Fagus sylvatica (Schäfer, Oren & Tenhunen 2000). Saliendra, Sperry & Comstock (1995) found no evidence for acclimation of either specific sapwood conductance or leaf area with increasing path length in Betula occidentalis, however, there was a more than twofold decline in whole-plant leaf specific conductance between juvenile and adult individuals. Apparently, the acclimation of leaf area with height is not sufficient to prevent leaf-specific hydraulic conductance from declining with height; it merely mitigates the reduction in leaf-specific hydraulic conductance (McDowell et al. 2002).

Does acclimation to increased hydraulic resistance affect forest growth?

  1. Top of page
  2. Summary
  3. Introduction
  4. Can vessel tapering compensate for height growth?
  5. Are the predictions of WBE confirmed by observations?
  6. Does acclimation to increased hydraulic resistance affect forest growth?
  7. Concluding remarks
  8. Acknowledgements
  9. References
  10. Appendix

A recent study on a moist-tropical Eucalyptus saligna plantation suggested that a decline in growth occurred, even though an acclimation of leaf area with height compensated for the hydraulic effect of height (Barnard & Ryan 2003). Direct and indirect evidence suggests that maintenance and growth respiration costs of the stem are not greater in taller trees (Ryan et al. 1992; Mencuccini & Grace 1996b), so increasing respiration cannot be invoked as an explanation. Instead, the hydraulic limitation theory might need to be extended. Even in the absence of any hydraulic limitation related to path length, tree height might affect leaf water potential through gravity (Woodruff, Bond & Meinzer 2004). Osmotic adjustments as a response to loss of turgor (Woodruff et al. 2004), or increased leaf construction costs to maintain lower minimum leaf water potentials (Barnard & Ryan 2003; Koch et al. 2004) in taller trees, could only partly offset reductions in growth, as these adaptations would divert resource allocation from photosynthesis. Similarly, increased below-ground C allocation and decreased leaf-to-sapwood area ratios to minimize the increase of total plant hydraulic resistance with height (Magnani, Mencuccini & Grace 2000) could mitigate the effect of tree height on stomatal conductance, but only at increased costs per unit foliage, thereby ultimately limiting tree growth.

Concluding remarks

  1. Top of page
  2. Summary
  3. Introduction
  4. Can vessel tapering compensate for height growth?
  5. Are the predictions of WBE confirmed by observations?
  6. Does acclimation to increased hydraulic resistance affect forest growth?
  7. Concluding remarks
  8. Acknowledgements
  9. References
  10. Appendix

The WBE model shows that tapering could theoretically compensate for the effect of tree height on hydraulic resistance, and therefore limit the hydraulic effect of tree height on tree growth. The predicted scaling of hydraulic conductance matches observed scaling on a segment scale, but not at more aggregated levels. Thus, either the WBE model lacks an important process, or one or more of its assumptions (e.g. conservation of total tube number; uniformity of tapering; or the volume-filling and area-preserving branching network) are invalid. However, the data available are insufficient to evaluate all the assumptions underlying the WBE model, even though they indicate that the assumption of constant tube numbers is too strong. Nevertheless, the data do not support the prediction of the WBE model that whole-tree hydraulic conductance is generally independent of height. Tapering cannot exclude that the hydraulic limitation hypothesis describes the principal mechanism for the observed decline in stand productivity with age. Instead, the hydraulic limitation theory should be extended to account for hydraulic acclimation and the associated consequences for gross primary production and C allocation. For the mechanistic modelling of forest development with age, it is essential to understand tree hydraulic architecture and its implications for tree growth.

Acknowledgements

  1. Top of page
  2. Summary
  3. Introduction
  4. Can vessel tapering compensate for height growth?
  5. Are the predictions of WBE confirmed by observations?
  6. Does acclimation to increased hydraulic resistance affect forest growth?
  7. Concluding remarks
  8. Acknowledgements
  9. References
  10. Appendix

This review was supported by the HSB programme of the Federal State of Brandenburg, Germany (AZ: 24-04/323; 200). The author is grateful to Stephen Sitch, I. Colin Prentice and two anonymous reviewers for helpful comments on previous versions of this manuscript.

References

  1. Top of page
  2. Summary
  3. Introduction
  4. Can vessel tapering compensate for height growth?
  5. Are the predictions of WBE confirmed by observations?
  6. Does acclimation to increased hydraulic resistance affect forest growth?
  7. Concluding remarks
  8. Acknowledgements
  9. References
  10. Appendix
  • Barnard, H.R. & Ryan, M.G. (2003) A test of the hydraulic limitation hypothesis in fast-growing Eucalyptus saligna. Plant, Cell & Environment 26, 12351245.
  • Becker, P., Gribben, R.J. (2001) Estimation of conduit taper for the hydraulic resistance model of West et al. Tree Physiology 21, 697700.
  • Becker, P., Gribben, R.J. & Lim, C.M. (2000a) Tapered conduits can buffer hydraulic conductance from path-length effects. Tree Physiology 20, 965967.
  • Becker, P., Meinzer, F.C. & Wullschleger, S.D. (2000b) Hydraulic limitation of tree height: a critique. Functional Ecology 14, 411.
  • Bond, B.J. & Ryan, M.G. (2000) Comment on ‘Hydraulic limitation of tree height: a critique’ by Becker, Meintzer and Wullschleger. Functional Ecology 14, 137140.
  • Dodds, P.S., Rothman, D.H. & Weitz, J.S. (2001) Re-examination of the ‘3/4-law’ of metabolism. Journal of Theoretical Biology 209, 927.
  • Enquist, B.J. (2002) Universal scaling in tree and vascular plant allometry: toward a general quantitative theory linking plant form and function from cell to ecosystems. Tree Physiology 22, 10451064.
  • James, S.A., Meinzer, F.C., Goldstein, G. et al. (2003) Axial and radial water transport and internal water storage in tropical forest canopy trees. Oecologia 134, 3745.
  • Jarvis, P.G. (1975) Water transfer in plants. Heat Mass Transfer in the Plant Environment, Part 1 (eds D.A.De Vries & N.G.Afgan), pp. 369394. Scripta Book Co., Washington, DC, USA.
  • Koch, G.W., Sillett, S.C., Jennings, G.M. & Davis, S.D. (2004) The limits to tree height. Nature 428, 851854.
  • Magnani, F., Mencuccini, M. & Grace, J. (2000) Age-related decline in stand productivity: the role of structural acclimation under hydraulic constraints. Plant, Cell & Environment 23, 251263.
  • McCulloh, K.A., Sperry, J.S. & Adler, F.R. (2003) Water transport in plants obeys Murray's law. Nature 421, 939942.
  • McDowell, N., Barnard, H., Bond, B.J. et al. (2002) The relationship between tree height and leaf area : sapwood area ratio. Oecologia 132, 1220.
  • Mencuccini, M. (2002) Hydraulic constraints in the functional scaling of trees. Tree Physiology 22, 553565.
  • Mencuccini, M. & Grace, J. (1996a) Developmental patterns in aboveground hydraulic conductance in a Scots pine (Pinus sylvestris L.) age sequence. Plant, Cell & Environment 19, 939948.
  • Mencuccini, M. & Grace, J. (1996b) Hydraulic conductance, light inception and needle nutrient concentration in Scots pine stands and their relation with net primary productivity. Tree Physiology 16, 459468.
  • Mencuccini, M. & Magnani, F. (2000) Comment on ‘Hydraulic limitation of tree height: a critique’ by Becker, Meintzer and Wullschleger. Functional Ecology 14, 135137.
  • Nobel, P.S. (1983) Biophysical Plant Physiology and Ecology. W.H. Freeman, San Francisco, CA, USA.
  • Pothier, D., Margolis, H.A. & Waring, R.H. (1989) Patterns of change in saturated sapwood permeability and sapwood conductance with stand development. Canadian Journal of Forest Research 19, 432439.
  • Ryan, M.G. & Waring, R.H. (1992) Maintenance respiration and stand development in a subalpine lodgepole pine forest. Ecology 73, 21002108.
  • Ryan, M.G. & Yoder, B.J. (1997) Hydraulic limits to tree height and tree growth. Bioscience 47, 235242.
  • Ryan, M.G., Gower, S.T., Hubbard, R.M. et al. (1995) Woody tissue maintenance respiration of four conifers in contrasting climates. Oecologia 101, 133140.
  • Ryan, M.G., Binkley, D. & Fownes, J.H. (1997) Age-related decline in forest productivity: pattern and process. Advances in Ecological Research 27, 213262.
  • Ryan, M.G., Bond, B.J., Law, B.E. et al. (2000) Transpiration and whole-tree conductance in ponderosa pine trees of different heights. Oecologia 124, 553560.
  • Saliendra, N.Z., Sperry, J.S. & Comstock, J.P. (1995) Influence of leaf water status on stomatal response to humidity, hydraulic conductance, and soil drought in Betula occidentalis. Planta 196, 357366.
  • Schäfer, K., Oren, R. & Tenhunen, J. (2000) The effect of height on crown level stomatal conductance. Plant, Cell & Environment 23, 365375.
  • Shinozaki, K., Yoda, K., Hozumi, K. & Kira, T. (1964a) A quantitative analysis of the plant form – the pipe model theory. II. Further evidence of the theory and its application in forest ecology. Japanese Journal of Ecology 14, 133139.
  • Shinozaki, K., Yoda, K., Hozumi, K. & Kira, T. (1964b) A quantitative analysis of the plant form – the pipe model theory. Japanese Journal of Ecology 14, 98104.
  • Tyree, M.T. & Sperry, J.S. (1988) Do woody plants operate near the point of catastrophic xylem dysfunction caused by dynamic water stress? Plant Physiology 88, 574580.
  • West, G.B., Brown, J.H. & Enquist, B.J. (1997) A General model for the origin of allometric scaling laws in biology. Science 276, 122126.
  • West, G., Brown, J. & Enquist, B. (1999) A general model for the structure and allometry of plant vascular systems. Nature 400, 664667.
  • Whitehead, D. & Hinckley, T.M. (1991) Models of water flux through forest stands: critical leaf and stand parameters. Tree Physiology 9, 3557.
  • Whitehead, D., Edwards, W.R.N. & Jarvis, P.G. (1984) Conducting sapwood area, foliage area, and permeability in mature trees of Picea sitchensis and Pinus cordata. Canadian Journal of Forest Research – Revue Canadienne de Recherche Forestiere 14, 940947.
  • Woodruff, D.R., Bond, B.J. & Meinzer, F.C. (2004) Does turgor limit growth in tall trees? Plant, Cell & Environment 27, 229236.
  • Yang, S. & Tyree, M.T. (1993) Hydraulic resistance in Acer saccharum shoots and its influence on leaf water potential and transpiration. Tree Physiology 12, 231242.
  • Yang, S. & Tyree, M.T. (1994) Hydraulic architecture of Acer saccharum and Acer rubrum: comparison of branches to whole trees and the contribution of leaves to hydraulic resistance. Journal of Experimental Botany 45, 179186.
  • Yoda, K., Shinozaki, K., Ogawa, H., Hozumi, K. & Kira, T. (1965) Estimation of total amount of respiration from woody organs of trees and forest communities. Journal of Biology 16, 1526.
  • Yoder, B.J., Ryan, M.G., Waring, R.H., Schoettle, A.W. & Kaufmann, M.R. (1994) Evidence of reduced photosynthetic rates in old trees. Forest Science 40, 513527.
  • Zimmermann, M.H. (1978) Hydraulic architecture of some diffuse-porous trees. Canadian Journal of Forest Research 56, 22862295.

Appendix

  1. Top of page
  2. Summary
  3. Introduction
  4. Can vessel tapering compensate for height growth?
  5. Are the predictions of WBE confirmed by observations?
  6. Does acclimation to increased hydraulic resistance affect forest growth?
  7. Concluding remarks
  8. Acknowledgements
  9. References
  10. Appendix

The initial equations from the West et al. (1999) model (WBE) describe a fractal network for the adjacent levels of branches k and k + 1:

  • image((eqn A1) )

where n is the number of daughter branches, and α, inline image and β are the scaling coefficients for branch diameter r, tube radius a and segment length l, respectively. As the WBE assumes invariant petiole conduit diameter and length, it is more convenient to invert the equations in equation A1. The hydraulic resistance Z of a single tube j within a branch segment k can be approximated by the Hagen–Poiseuille formula, which describes capillary flow. This formula neglects the substantial contribution of pit membrane resistance to total resistance, but this does not affect relative comparisons of resistance (Nobel 1983; Becker et al. 2000a).

  • image((eqn A2) )

where η is viscosity. Assuming that the number of total tubes is preserved, the ratio of resistances i between adjacent levels is given by:

  • image((eqn A3) )