Universal scaling in tree and vascular plant allometry: Toward a general quantitative theory linking plant form and function from cells to ecosystems

Tree Physiology

Volumn 22, Issue 15-16, 2002, Pages 1045-1064

  Enquist, Brian J ^a  

^a UNIVERSITY OF ARIZONA   (United States)

Author keywords

Diversity; Energy dissipation; Evolution; Fractal networks; Metabolism; Space filling

Indexed keywords

ALLOMETRY; PLANT COMMUNITY; SCALE EFFECT; THEORY; VASCULAR PLANT;

TRACHEOPHYTA;

EID: 0036847811     PISSN: 0829318X     EISSN: None     Source Type: Journal    
DOI: 10.1093/treephys/22.15-16.1045     Document Type: Conference Paper

References (162)

1. Agusti, S., C.M. Duarte and J. Kalff. 1987. Algal size and the maximum density and biomass of phytoplankton. Limnol. Oceanogr. 32:983-986.
2. Aloni, R., J.D. Alexander and M.T. Tyree. 1997. Natural and experimentally altered hydraulic architecture of branch junctions in Acer saccharum Marsh. and Quercus velutina Lam. trees. Trees 11:255-264.
3. Aono, M. and L.K. Tosiyasu. 1984. Botanical tree image generation. I.E.E.E. Comput. Graph. 27:10-34.
4. Apple, M., K. Tiekotter, M. Snow, J. Young, A. Soeldner, D. Phillips, D. Tingey and B.J. Bond. 2002. Needle anatomy changes with increasing tree age in Douglas-fir. Tree Physiol. 22:181-188.
5. Barton-Wright, E.C. 1932. Recent Advances in Botany. J & A. Churchill, London, 287 p.
6. Basinger, J.F., D.R. Greenwood and T. Sweda. 1994. Early tertiary vegetation of arctic Canada and its relevance to paleoclimatic interpretation. In Cenozoic Plants and Climates of the Arctic. Eds. M.C. Boulter and H.C. Fisher. Springer-Verlag, Berlin, pp 175-198.
7. Becker, P., R.J. Gribben and C.M. Lim. 2000. Tapered conduits can buffer hydraulic conductance from path-length effects. Tree Physiol. 20:965-967.
8. Berninger, F. and E. Nikinmaa. 1997. Implications of varying pipe model relationships on Scots pine growth in different climates. Funct. Ecol. 11:146-156.
9. Bernston, G.M. and P. Stoll. 1997. Correcting for finite spatial scales of self-similarity when calculating the fractal dimensions of real-world structures. Proc. R. Soc. Lond. Ser. B Biol. Sci. 64: 1531-1537.
10. Bertram, J.E.A. 1989. Size-dependent differential scaling in branches: The mechanical design of trees revisited. Trees 4:241-253.
11. Borchert, R. and N.A. Slade. 1981. Bifurcation ratios and the adaptive geometry of trees. Bot. Gaz. 142:394-401.
12. Brown, J.H. 1995. Macroecology. University of Chicago Press, Chicago, 269 p.
13. Brown, J.H., G.B. West and B.J. Enquist. 2000. Scaling in biology: Patterns and processes, causes and consequences. In Scaling in Biology. Eds. J.H. Brown and G.B. West. Oxford University Press, Oxford, pp 167-198.
14. Brown, S. 1997. Estimating Biomass and Biomass Change of Tropical Forests. Food and Agricultural Organization of the United Nations Forestry Paper 134, 55 p.
15. Calder, W.A.I. 1984. Size, Function and Life History. Harvard University Press, Cambridge, 431 p.
16. Cannell, M.G.R. 1982. World Forest Biomass and Primary Production Data. Academic Press, London, 391 p.
17. Carlquist, S. 1988. Comparative Wood Anatomy: Systematic, Ecological, and Evolutionary Aspects of Dicotyledon Wood. Springer-Verlag, New York, 436 p.
18. Chambers, J.Q., J. dos Santos, R.J. Ribeiro and N. Higuchi. 2001. Tree damage, allometric relationships, and above-ground net primary production in central Amazon forest. For. Ecol. Manage. 152:73-84.
19. Chapin, F.S., III. 1993. Functional role of growth forms in ecosystem and global processes. In Scaling Physiological Processes: Leaf to Globe. Eds. J.R. Ehleringer and C.B. Field. Academic Press, New York, pp 287-308.
20. Charnov, E.L. 1993. Life History Invariants: Some Explorations of Symmetry in Evolutionary Ecology. Oxford University Press, Oxford, 167 p.
21. Comstock, J.P. and J.S. Sperry. 2000. Theoretical considerations of optimal conduit length for water transport in vascular plants. New Phytol. 148:195-218.
22. Damuth, J. 1981. Population density and body size in mammals. Nature 290:699-700.
23. Damuth, J. 1987. Interspecific allometry of population density in mammals and other animals: The independence of body mass and population energy-use. Biol. J. Linn. Soc. 31:193-246.
24. Damuth, J. 2001. Scaling of growth: Plants and animals are not so different. Proc. Natl. Acad. Sci. 98:2113-2114.
25. De Liocourt, F. 1898. De l'amenagement des sapinieres. Bull. Soc. For. Franche-Comte Belfort, Besancon, pp 396-409.
26. Dewar, R.C. 1993. A mechanistic analysis of self-thinning in terms of carbon balance of trees. Ann. Bot. 71:147-159.
27. Dewar, R.C., B.E. Medlyn and R.R. McMurtrie. 1998. A mechanistic analysis of light and carbon use efficiencies. Plant Cell Environ. 21:573-588.
28. Ehleringer, J.R. and C.B. Field. 1993. Scaling Physiological Processes: Leaf to Globe. Academic Press, New York, 388 p.
29. Ellison, A.M. 1987. Density-dependent dynamics of Salicornia europaea monocultures. Ecology 68:737-741.
30. Ellison, A.M. and K.J. Niklas. 1988. Branching patterns of Salicornia europaea (Chenopodiaceae) at different successional stages: A comparison of theoretical and real plants. Am. J. Bot. 75:501-512.
31. Ellison, A.M., K.J. Niklas and S. Shumway. 1993. Xylem vascular anatomy and water transport of Salicornia europaea. Aquatic Bot. 45:325-339.
32. Enquist, B.J. 2003. Scaling the macroecological and evolutionary implications of size and metabolism within and across plant taxa. In Macroecology: Reconciling Divergent Perspectives on Large Scale Ecological Patterns. Eds. T. Blackburn and K. Gaston. Oxford University Press, Oxford. In press.
33. Enquist, B.J. and K.J. Niklas. 2001. Invariant scaling relations across tree-dominated communities. Nature 410:655-660.
34. Enquist, B.J. and K.J. Niklas. 2002. Global allocation rules for patterns of biomass partitioning in seed plants. Science 295:1517-1520.
35. Enquist, B.J., J.H. Brown and G.B. West. 1998. Allometric scaling of plant energetics and population density. Nature 395:163-165.
36. Enquist, B.J., G.B. West, E.L. Charnov and J.H. Brown. 1999. Allometric scaling of production and life-history variation in vascular plants. Nature 401:907-911.
37. Enquist, B.J., G.B. West and J.H. Brown. 2000. Quarter-power scaling in vascular plants: Functional basis and ecological consequences. In Scaling in Biology. Eds. J.H. Brown and G.B. West. Oxford University Press, Oxford, pp 167-198.
38. Enquist, B.J., J.P. Haskell, B. Tiffney and K.J. Niklas. 2001. The Evolution of Plant Communities: Biodiversity and Community Structure Through Time. Volume 4. Ed. S.A. Levin. Academic Press, New York, pp 631-644.
39. Eshel, A. 1998. On the fractal dimensions of a root system. Plant Cell Environ. 21:247-251.
40. Ewers, F.W. and M.H. Zimmermann. 1984. The hydraulic architecture of eastern hemlock (Tsuga canadensis). Can. J. Bot. 62:940-946.
41. Farnsworth, K.D. and K.J. Niklas. 1995. Theories of opitimization, form and function in branching architecture in plants. Funct. Ecol. 9:355-363.
42. Fitter, A.H. and T.R. Strickland. 1992. Fractal characterization of root system architecture. Funct. Ecol. 6:632-635.
43. Gentry, A.H. 1988. Changes in plant community diversity and floristic composition on environmental and geographic gradients. Ann. Mo. Bot. Gard. 75:1-34.
44. Gentry, A.H. 1995. Patterns of diversity and floristic composition in neotropical montane forests. In Biodiversity and Conservation of Neotropical and Montane Forests. Eds. S.P. Churchill, H. Balslev, E. Forero and J.L. Luteyn. The New York Botanical Garden, New York, pp 103-126.
45. Givnish, T. 1982. On the adaptive significance of leaf height in forest herbs. Am. Nat. 120:353-381.
46. Givnish, T. 1987. Comparative studies of leaf form: Assessing the relative roles of selective pressures and phylogenetic constraints. New Phytol. 106:131-160.
47. Gorham, E. 1979. Shoot height, weight and standing crop in relation to density in monospecific plant stands. Nature 279:148-150.
48. Grace, J. 1997. Plant water relations. In Plant Ecology. Ed. M.J. Crawley. Blackwell Science, Cambridge, pp 28-50.
49. Greenhill, A.G. 1881. Determination of the greatest height consistent with stability that a verticle pole or mast can be made, and the greatest height to which a tree of given proportions can grow. Proc. Camb. Philos. Soc. 4:65-73.
50. Grime, J.P. and R. Hunt. 1975. Relative growth-rate: Its range and adaptive significance in a local flora. J. Ecol. 63:393-422.
51. Hacke, U.G. and J.S. Sperry. 2001. Functional and ecological xylem anatomy. Perspect. Plant Ecol. Evol. Syst. 4:97-115.
52. Hacke U.G., J.S. Sperry, W.T. Pockman, S.D. Davis, K.A. McCulloch. 2001. Trends in wood density and structure are linked to prevention of xylem implosion by negative pressure. Oecologia 126:457-461.
53. Halle, F., R.A.A. Oldemann and P.B. Tomlinson. 1978. Tropical Trees and Forests: An Architectural Analysis. Springer-Verlag, Berlin, pp 209-221.
54. Harper, J. 1967. A Darwinian approach to plant ecology. J. Ecol. 55:247-270.
55. Harper, J. 1977. Population Biology of Plants. Academic Press, New York, 892 p.
56. Honda, H. 1971. Description of the form of the tree-like body: Effects of branching angle and the branch length on the shape of the tree-like body. J. Theor. Biol. 31:331-338.
57. Horn, H.S. 1971. The Adaptive Geometry of Trees. Princeton University Press, Princeton, 144 p.
58. Horn, H.S. 2000. Twigs, trees, and the dynamics of carbon in the landscape. In Scaling in Biology. Eds. J.H. Brown and G.B. West. Oxford University Press, Oxford, pp 199-220.
59. Huber, B. 1928. Weitere quantitative untersuchungen uber sas wasserleitungssystem der pflanzen. Jb. Wiss. Bot. 67:877-959.
60. Huber, B. 1932. Observation and measurement of plant sap streams. Ber. Dtsh. Bot. Ges. 50:89-109.
61. Huber, B. and E. Schmidt. 1936. Weitere thermo-elektrische untersuchungen uber den transpirationsstrom der baume. Tharandt Forst. Jb. 87:369-412.
62. Hutchings, M. 1983. Ecology's law in search of a theory. New Sci. 98:765-767.
63. Huxley, J.S. 1932. Problems of Relative Growth. Methuea, Methuea.
64. James, S.A., M.J. Clearwater, F.C. Meinzer et al. 2002. Heat dissipation sensors of variable length for the measurement of sap flow in trees with deep sapwood. Tree Physiol. 22:277-283.
65. Jones, H.G. 1978. Modeling diurnal trends of leaf water potential in transpiring wheat. J. Appl. Ecol. 15:613-626.
66. Jones, H.G. 1992. Plants and Microclimate: A Quantitative Approach to Environmental Plant Physiology. Cambridge University Press, Cambridge, 428 p.
67. King, D.A. and O.L. Louks. 1978. The theory of tree bole and branch form. Radiat. Environ. Biophys. 15:141-165.
68. Kuuluvainen, T., D.G. Sprugel and J.R. Brooks. 1996. Hydraulic architecture and structure of Abies lasiocarpa seedlings in three alpine meadows of different moisture status in the eastern Olympic mountains, Washington, U.S.A. Arct. Alp. Res. 28:60-64.
69. Leopold, L.B. 1971. Trees and streams: The efficiency of branching patterns. J. Theor. Biol. 31:339-354.
70. Lev-Yadun, S. and R. Aloni. 1990. Vascular differentiation in branch junctions of trees: Circular patterns and functional significance. Trees 4:49-54.
71. Levin, S. 1992. The problem of pattern and scale in ecology. Ecology 73:1943-1967.
72. Lonsdale, W.M. 1990. The self-thinning rule: Dead or alive? Ecology 71:1373-1388.
73. Mandelbrot, B.B. 1977. The Fractal Geometry of Nature. W.H. Freeman, New York, 468 p.
74. McMahon, T.A. 1973. Size and shape in biology. Science 179:1201-1204.
75. McMahon, T.A. and J.T. Bonner. 1983. On Size and Life. Scientific American Library, New York, 255 p.
76. McMahon, T.A. and R.E. Kronauer. 1976. Tree Structures: Deducing the principle of mechanical design. J. Theor. Biol. 59:443-466.
77. Meinzer, F.C., G. Goldstein, J.L. Andrade. 2001. Regulation of water flux through tropical forest canopy trees: Do universal rules apply? Tree Physiol. 21:19-26.
78. Miyanishi, K., A.R. Hoy and P.B. cavers. 1979. A generalized law of self-thinning in plant populations. J. Theor. Biol. 78:439-442.
79. Morgan, J. and M.G.R. Cannell. 1988. Support costs of different branch designs: Effects of position, number, angle and deflection of laterals. Tree Physiol. 4:303-313.
80. Morse, D.R., J.H. Lawton, J.H. Dodson and M.M. Williamson. 1985. Fractal dimension of vegetation and the distribution of arthropod body lengths. Nature 314:731-733.
81. Murray, C.D. 1927. A relationship between circumference and weight in trees and its bearing on branching angles. J. Gen. Physiol. 10:725-739.
82. Nielson, K.L., J.P. Lynch and H.N. Weiss. 1997. Fractal geometry of bean root systems: Correlations between spatial and fractal dimension. Am. J. Bot. 84:26-33.
83. Niklas, K.J. 1984. Size-related changes in the primary xylem anatomy of some early tracheophytes. Paleobiology 10:487-506.
84. Niklas, K.J. 1988. The role of phyllotactic pattern as a developmental constraint on the interception of light by leaf surfaces. Evolution 40:1-16.
85. Niklas, K.J. 1992. Plant Biomechanics: An Engineering Approach to plant Form and Function. University of Chicago Press, Chicago, 607 p.
86. Niklas, K.J. 1993. The allometry of plant reproductive biomass and stem diameter. Am. J. Bot. 80:461-467.
87. Niklas, K.J. 1994a. Morphological evolution through complex domains of fitness. Proc. Natl. Acad. Sci. 91:6772-6779.
88. Niklas, K.J. 1994b. Plant Allometry: The Scaling of Form and Process. University of Chicago Press, Chicago, 395 p.
89. Niklas, K.J. 1994c. Size-dependent variations in plant growth rates and the "3/4-power rule". Am. J. Bot. 81:134-145.
90. Niklas, K.J. 1997a. Effects of hypothetical developmental barriers and abrupt Changes on adaptive walks in a computer generated domain for early vascular land plants. Paleobiology 23:63-67.
91. Niklas, K.J. 1997b. Size- and age-dependent variation in the properties of sap- and heartwood in black locust (Robinia pseudoacacia L.) Ann. Bot. 79:473-478.
92. Niklas, K.J. and B.J. Enquist. 2001. Invariant scaling relationships for interspecific plant biomass production rates and body size. Proc. Natl. Acad. Sci. 98:2922-2927.
93. Niklas, K.J. and B.J. Enquist. 2002. On the vegetative biomass partitioning of seed plant leaves, stems, and roots. Am. Nat. 159:482-497.
94. Niklas, K.J. and V. Kerchner. 1984. Mechanical and photosynthetic constraints on the evolution of plant shape. Paleobiology 10:79-101.
95. Norberg, R.A. 1988. Theory of growth geometry of plants and self-thinning of plant populations: Geometric similarity, elastic similarity, and different growth modes of plant parts. Am. Nat. 131:220-256.
96. Noshiro, S. and M. Suzuki. 2001. Ontogenetic wood anatomy of tree and subtree species of Nepalese Rhododendron (Ericaceae) and characterization of shrub species. Am. J. Bot. 88:560-569.
97. Osawa, A. and S. Suigita. 1989. The self-thinning rule: Another interpretation of Weller's results. Ecology 70:279-283.
98. Patino, S., M.T. Tyree and H. Herre. 1995. Comparison of hydraulic architecture of wood plants of differing phylogeny and growth form with special reference to free standing and hemi-epiphytic Ficus species from Panama. New Phytol. 129:125-134.
99. Pearsall, W.H. 1927. Growth studies. VI. On the relative sizes of growing plant organs. Ann. Bot. 41:549-556.
100. Peters, R.H. 1983. The Ecological Implications of Body Size. Cambridge University Press, Cambridge, 329 p.
101. Petraitis, P.S. 1995. Use of averages versus total biomass in self-thinning relationships. Ecology 76:656-658.
102. Pole, M. 1999. Structure of a near-polar latitude forest from the New Zealand Jurassic. Palaeogeogr. Palaeoclimatol. Palaeoecol. 147:121-139.
103. Rashevsky, N. 1973. The principle of adequate design. In Foundations of Mathematical Biology. Vol. III. Ed. R. Rosen. Academic Press, New York, pp 143-175.
104. Raven, J.A. 1993. The evolution of vascular plants in relation to quantitative functioning of dead water conducting cells and stomata. Biol. Rev. 68:337-363.
105. Raven, J.A. and L.L. Handley. 1987. Transport processes and water relations. New Phytol. 106:217-233.
106. Reiss, M. 1989. Plant resource allocation. Trends Ecol. Evol. 4:379.
107. Richter, J.P. 1970. The Notebooks of Leonardo da Vinci 1452-1519. Dover Press, New York, 432 p.
108. Robichaud, E. and I.R. Methven. 1992. The applicability of the pipe model theory for the prediction of foliage biomass in trees from natural, untreated black spruce stands. Can. J. For. Res. 22:1118-1123.
109. Rogers, R. and T.M. Hinckley. 1979. Foliar weight and area related to current sapwood area in oak. For. Sci. 25:298-303.
110. Rosenzweig, M.L. 1968. Net primary productivity of terrestrial communities: Prediction from climatological data. Am. Nat. 102:67-74.
111. Ryan, M.G. and B.J. Yoder. 1997. Hydraulic limits to tree height and tree growth: What keeps trees from growing beyond a certain height? BioScience 47:235-242.
112. Sakurantani, T. 1981. A heat balance method for measuring water flux in the stem of intact plants. J. Agric. Meteorol. 37:9-17.
113. Schmidt-Nielsen, K. 1984. Scaling: Why is Animal Size so Important? Cambridge University Press, Cambridge, 241 p.
114. Schulte, P.J. and D.G. Costa. 1996. A mathematical description of water flow through plant tissues. J. Theor. Biol. 180:61-70.
115. Schulze, E.-D., J. Čermák, R. Matyssek, M. Penka, R. Zimmermann, F. Vasicek, W. Gries and J. Kučera. 1985. Canopy transpiration and water fluxes in the xylem of the trunk of Larix and Picea trees - A comparison of xylem flow, porometer and cuvette measurements. Oecologia 66:475-483.
116. Schulze, E.-D., F.M. Kelliher, C. Korner, J. Lloyd and R. Leuning. 1994. Relationships among maximal stomatal conductance, carbon assimilation rate, and plant nitrogen nutrition: A global ecology scaling exercise. Annu. Rev. Ecol. Syst. 25:629-660.
117. Shidei, T. and T. Kira. 1977. Primary Productivity of Japanese Forests: Productivity of Terrestrial Communities (JIBP Synthesis). University of Tokyo Press, Tokyo, 435 p.
118. Shinozaki, K., K. Yoda, K. Hozumi and T. Kira. 1964. A quantitative analysis of plant form - The pipe model theory. I. Basic analysis. Jpn. J. Ecol. 14:97-105.
119. Sinnott, E.W. 1921. The relation between body size and organ size in plants. Am. Nat. 55:385-403.
120. Smith, W.B. and G.J. Brand. 1983. Allometric Biomass Equations for 98 Species of Herbs, Shrubs, and Small Trees. Res. Note NC-299. St. Paul, MN: U.S. Department of Agriculture, Forest Service, North Central Forest Experiment Station, 8 p.
121. Smith, J.A.C., P.J. Schulte and P.S. Nobel. 1987. Water flow and water storage in Agave deserti: Osmotic implications of crassulacean acid metabolism. Plant Cell Environ. 10:639-648.
122. Sperry, J.S., N.N. Alder and S.E. Eastlack 1993. The effect of reduced hydraulic conductance on stomatal conductance and xylem cavitation. J. Exp. Bot. 44:1075-1082.
123. Stearns, S.C. 1992. The Evolution of Life Histories. Oxford University Press, Oxford, 248 p.
124. Stevens, G.C. 1987. Lianas as structural parasites: The Bursera simaruba example. Ecology 68:77-81.
125. Tatsumi, J.A., A. Yamauchi and Y. Kono. 1989. Fractal analysis of plant root systems. Ann. Bot. 64:499-503.
126. Thomas, S.C. 1996a. Asymptotic height as a predictor of growth and allometric characterizations in Malaysian rain forest trees. Am. J. Bot. 83:556-566.
127. Thomas, S.C. 1996b. Reproductive allometry in Malaysian rain forest trees: Biomechanics versus optimal allocation. Evol. Ecol. 10:517-530.
128. Thompson, D'A.W. 1942. On Growth and Form. Cambridge Univ. Press, Cambridge, 1116 p.
129. Tilman, D. 1988. Plant Strategies and the Dynamics and Structure of Plant Communities. Princeton University Press, Princeton, 376 p.
130. Tilman, D., C.L. Lehman and K.T. Thomson. 1997. Plant diversity and ecosystem productivity: Theoretical considerations. Proc. Natl. Acad. Sci. 94:1857-1861.
131. Turrell, F.M. 1961. Growth of the photosynthetic area of Citrus. Bot. Gaz. 122:284-298.
132. Tyree, M.T. 1988. A dynamic model for water flow in a single tree: Evidence that models must account for hydraulic architecture. Tree Physiol. 4:195-217.
133. Tyree, M.T. and J.D. Alexander. 1993. Hydraulic conductivity of branch junctions in three temperate tree species. Trees 7:156-159.
134. Tyree, M.T. and F.W. Ewers. 1991. The hydraulic architecture of trees and other woody plants. New Phytol. 119:345-360.
135. Tyree, M.T. and F.W. Ewers. 1996. Hydraulic architecture of woody tropical plants. In Tropical Forest Plant Ecophysiology. Eds. S.S. Mulkey, R.L. Chazdon and A.P. Smith. Chapman and Hall, New York, pp 217-243.
136. Tyree, M.T. and J.S. Sperry. 1988. Do woody plants operate near the point of catastrophic xylem disfunction caused by dynamic water stress? Answers from a model. Plant Physiol. 88:574-580.
137. Tyree, M.T., M.E.D. Graham, K.E. Cooper and L.J. Bazos. 1983. The hydraulic architecture of Thuja occidentalis. Can. J. Bot. 61:2105-2111.
138. Van den Honert, T.H. 1948. Water transport in plants as a catenary process. Discuss. Faraday Soc. 3:146-153.
139. Waring, R.H., P.E. Schroeder and R. Oren. 1982. Application of the pipe model theory to predict canopy leaf area. Can. J. For. Res. 12:556-560.
140. Weller, D.E. 1987. A re-evaluation of the -3/2 power rule of plant self-thinning. Ecol. Monogr. 57:23-43.
141. Weller, D.E. 1989. The interspecific size-density relationship among crowded plant stands and its implications for the -3/2 power rule of self-thinning. Am. Nat. 133:20-41.
142. Weller, D.E. 1990. Will the real self-thinning rule please stand up? - A reply to Osawa and Sugita. Ecology 71:1204-1207.
143. Weller, D.E. 1991. The self-thinning rule: Dead or unsupported? - A reply to Lonsdale. Ecology 72:747-750.
144. West, G.B., J.H. Brown and B.J. Enquist. 1997. A general model for the origin of allometric scaling laws in biology. Science 276:122-126.
145. West, G.B., J.H. Brown and B.J. Enquist. 1999a. A general model for the structure, function, and allometry of plant vascular systems. Nature 400:664-667.
146. West, G.B., J.H. Brown and B.J. Enquist. 1999b. The fourth dimension of life: Fractal geometry and allometric scaling of organisms. Science 284:1677-1679.
147. West, G.B., J.H. Brown and B.J. Enquist. 2001. A general model for ontogenetic growth. Nature 413:628-631.
148. Westoby, M. 1984. The self-thinning rule. Adv. Ecol. Res. 14:167-225.
149. White, J. 1981. The allometric interpretation of the self-thinning rule. J. Theor. Biol. 89:475-500.
150. White, J. 1985. The thinning rule and its application to mixtures of plant populations. In Studies on Plant Demography: A Festschrift for John L. Harper. Ed. J. White. Academic Press, New York, pp 291-309.
151. Whitney, G.G. 1976. The bifurcation ratio as an indicator of adaptive strategy in woody plant species. Bull. Torrey Bot. Club 103:67-72.
152. Whittaker, R.H. and G.M. Woodwell. 1968. Dimension and production relations of trees and shrubs in the brookhaven forest, New York. J. Ecol. 57:1-25.
153. Yang, S. and M.T. Tyree. 1993. Hydraulic resistance in Acer saccharum shoots and its influence on leaf water potential and transpiration. Tree Physiol. 12:231-242.
154. Yang, S. and M.T. Tyree. 1994. Hydraulic architecture of Acer saccharum and A. rubrum: Comparison of branches to whole trees and the contribution of leaves to hydraulic resistance. J. Exp. Bot. 45:179-186.
155. Yoda, K., T. Kira, H. Ogawa and K. Hozumi. 1963. Self-thinning in overcrowded pure stands under cultivated and natural conditions. J. Biol. Osaka City Univ. 14:107-129.
156. Yoda, K., K. Shinozaki, H. Ogawa, K. Hozumi and T. Kira. 1965. Estimation of the total amount of respiration in woody organs of trees and forest communities. J. Biol. Osaka City Univ. 16:15-26.
157. Zeide, B. 1987. Analysis of the 3/2 power law of self-thinning. For. Sci. 33:517-537.
158. Zeide, B. 1991. Self-thinning and stand density. For. Sci. 37:517-523.
159. Zimmermann, M.H. 1978a. Hydraulic architecture of some diffuseporous trees. Can. J. Bot. 56:2286-2295.
160. Zimmermann, M.H. 1978b. Structural requirements for optimal water conduction in tree stems. In Tropical Trees as Living Systems. Eds. P.B. Tomlinson and M.H. Zimmermann. Cambridge University Press, New York, pp 517-532.
161. Zimmermann, M.H. 1983. Xylem Structure and the Ascent of Sap. Springer-Verlag, Berlin, 143 p.
162. Zimmermann, M.H. and C.L. Brown. 1971. Trees: Structure and Function. Springer-Verlag, New York, 336 p.