Abstract
Experimental plots covering a 120 years’ observation period in unthinned, even-aged pure stands of common beech (Fagus sylvatica), Norway spruce (Picea abies), Scots pine (Pinus sylvestris), and common oak (Quercus Petraea) are used to scrutinize Reineke’s (1933) empirically derived stand density rule ( \(N \propto \bar d^{-1.605} \), N = tree number per unit area, \(\bar{d}\) = mean stem diameter), Yoda’s (1963) self-thinning law based on Euclidian geometry (\(\bar w \propto N^{- 3/2}, \) \(\bar w\) = mean biomass per tree), and basic assumptions of West, Brown and Enquist’s (1997, 1999) fractal scaling rules (\(w \propto d^{8/3}, \) \(\bar w \propto N^{-4/3}, \) w = biomass per tree, d = stem diameter). RMA and OLS regression provides observed allometric exponents, which are tested against the exponents, expected by the considered rules. Hope for a consistent scaling law fades away, as observed exponents significantly correspond with the considered rules only in a minority of cases: (1) exponent r of \(N \propto \bar d^r \) varies around Reineke’s constant −1.605, but is significantly different from r=−2, supposed by Euclidian or fractal scaling, (2) Exponent c of the self-thinning line \(\bar w \propto N^c \) roams roughly about the Euclidian scaling constant −3/2, (3) Exponent a of \(w \propto d^a \) tends to follow fractal scaling 8/3. The unique dataset’s evaluation displays that (4) scaling exponents and their oscillation are species-specific, (5) Euclidian scaling of one relation and fractal scaling of another are coupled, depending on species. Ecological implications of the results in respect to self-tolerance (common oak > Norway spruce > Scots pine > common beech) and efficiency of space occupation (common beech > Scots pine > Norway spruce > common oak) are stressed and severe consequences for assessing, regulating and scheduling stand density are discussed.
Similar content being viewed by others
References
Assmann E (1970) The principles of forest yield study. Pergamon Press Ltd, Oxford
Bazzaz FA, Grace J (1997) Plant resource allocation. Academic, San Diego
Bégin E, Bégin J, Bélanger L, Rivest L-P, Tremblay St (2001) Balsam fir self-thinning relationship and its constancy among different ecological regions. Can J For Res 31:950–959
Begon ME, Harper JL, Townsend CR (1998) Ökologie. Spektrum Akademischer Verlag, Heidelberg
Bohonak AJ (2002) RMA. Software for reduced major axis regression, v. 1.14b, San Diego University. http://www.bio.sdsu.edu/pub/andy/rma.html
Ducey MJ, Larson BC (1999) Accounting for bias and uncertainty in nonlinear stand density indices. For Sci 45(3):452–457
Eid T, Tuhus E (2001) Models for individual tree mortality in Norway. For Ecol Manage 154:69–84
Enquist BJ, Niklas KJ (2001) Invariant scaling relations across tree-dominated communities. Nature 410:655–660
Enquist BJ, Brown JH, West GB (1998) Allometric scaling of plant energetics and population density. Nature 395:163–165
Enquist BJ, West GB, Charnov EL, Brown JH (1999) Allometric scaling of production and life-history variation in vascular plants. Nature 401:907–911
Foerster W (1990) Zusammenfassende ertragskundliche Auswertung der Kiefern-Düngungsversuchsflächen in Bayern. Forstl Forschungsberichte München 105:1–328
Foerster W (1993) Der Buchen-Durchforstungsversuch Mittelsinn 025. Allgemeine Forstzeitschrift 48:268–270
Franz F, Röhle H, Meyer F (1993) Wachstumsgang und Ertragsleistung der Buche. Allgemeine Forstzeitschrift 48:262–267
Gadow v K (1986) Observation on self-thinning in pine plantations. South African J Sci 82:364–368
Grote R, Schuck J, Block J, Pretzsch H (2003) Oberirdische holzige Biomasse in Kiefern-/Buchen- und Eichen-/Buchen-Mischbeständen. Forstw Cbl 122:287–301
Harper JL (1977) Population biology of plants. Academic, London New York
Kennel R (1972) Die Buchendurchforstungsversuche in Bayern von 1870 bis 1970. Forstl Forschungsberichte München 7:1–264
Kira T, Ogawa H, Sakazaki N (1953) Intraspecific competition among higher plants, I. Competition-yield-density interrelationship in regularly dispersed populations J Inst Polytech (Osaka City University) Ser D:1–16
Körner Ch (2002) Ökologie. In: Sitte P, Weiler EW, Kadereit JW, Bresinsky A, Körner Ch (eds) Strasburger Lehrbuch für Botanik, 35th edn. Spektrum Akademischer Verlag, Heidelberg Berlin, pp 886–1043
Kozlowski J, Konarzewski M (2004) Is West, Brown and Enquist’s model of allometric scaling mathematically correct and biologically relevant? Funct Ecol 18:283–289
Kramer H, Helms JA (1985) Zur Verwendung und Aussagefähigkeit von Bestandesdichteindizes bei Douglasie. Forstw Cbl 104:36–49
Küsters E (2001) Wachstumstrends der Kiefer in Bayern. PhD thesis, Wissenschaftszentrum Weihenstephan, Technische Universität München
Long JN, Smith FW (1984) Relation between size and density in developing stands: a description and possible mechanisms. For Ecol Manage 7:191–206
Mayer R (1958) Kronengröße und Zuwachsleistung der Traubeneiche auf süddeutschen Standorten. Allg Forst- u Jgdztg 129:105–114, 151–201
Niklas KJ (1994) Plant Allometry. University of Chicago Press, Chicago
Niklas KJ, Midgley JJ, Enquist BJ (2003) A general model for mass–growth–density relations across tree-dominated communities. Evol Ecol Res 5:459–468
Oliver CD, Larson BC (1990) Forest stand dynamics biological resource management series. McGraw-Hill, New York
Pittman SD, Turnblom EC (2003) A study of self-thinning using coupled loometric equations: implications for costal Douglas-fir stand dynamics. Can J For Res 33:1161–1669
Prairie YT, Bird DF (1989) Some misconceptions about the spurious correlation problem in the ecological literature. Oecologia 81:285–288
Pretzsch H (1985) Wachstumsmerkmale süddeutscher Kiefernbestände in den letzten 25 Jahren. Forstl Forschungsberichte München 65:1–183
Pretzsch H (2002) A unified law of spatial allometry for woody and herbaceous plants. Plant Biol 4:159–166
Pretzsch H, Biber P (2004) A re-evaluation of Reineke’s rule and stand density index. For Sci (accepted)
Pretzsch H, Utschig H (2000) Wachstumstrends der Fichte in Bayern. Mitt Bay Staatsforstverw 49:1–170
Puettmann KJ, Hibbs DE, Hann DW (1992) The dynamics of mixed stands of Alnus rubra and Pseudotsuga menziesii: extension of size-density analysis to species mixtures. J Ecol 80(3):449–458
Puettmann KJ, Hann DW, Hibbs DE (1993) Evaluation of the size-density relationship for pure red elder and Douglas-fir stands. For Sci 37:574–592
Reineke LH (1933) Perfecting a stand density index for even-aged forests. J Agric Res 46:627–638
Roderick ML, Barnes B (2004) Self-thinning of plant populations from a dynamic viewpoint. Funct Ecol 18:197–203
Röhle H (1994) Zum Wachstum der Fichte auf Hochleistungsstandorten in Südbayern. Habil-schrift, Universität München, Freising
Sackville Hamilton NR, Matthew C, Lemaire G (1995) In defence of the −3/2 boundary rule: a re-evaluation of self-thinning concepts and status. An Bot 76:569–577
Sterba H (1981) Natürlicher Bestockungsgrad und Reinekes SDI. Centralbl f d ges Forstw 98:101–116
Sterba H (1987) Estimating potential density from thinning experiments and inventory data. For Sci 33:1022–1034
Sterba H, Monserud RA (1993) The maximum density concept appled ton uneven-aged mixed stands. For Sci 39:432–452
Stoll P, Weiner J, Muller-Landau H, Müller E, Hara T (2002) Size symmetry of competition alters biomass–density relationships. Proc R Soc Lond B Biol Sci 269:2191–2195
Trendelenburg R, Mayer-Wegelin H (1955) Das Holz als Rohstoff. Hanser Verlag, Meunchen
Utschig H, Pretzsch H (2001) Der Eichen-Durchforstungsversuch Waldleiningen 88. Forstw Cbl 120:90–113
Verein Deutscher Forstlicher Versuchsanstalten (1902) Beratungen der vom Vereine Deutscher Forstlicher Versuchsanstalten eingesetzten Kommission zur Feststellung des neuen Arbeitsplanes für Durchforstungs- und Lichtungsversuche. Allg Forst- u Jgdztg 78:180–184
Weller DE (1987) A reevaluation of the −3/2 power rule of plant self-thinning. Ecol Monogr 57:23–43
Weller DE (1990) Will the real self-thinning rule please stand up? a reply to Osawa and Sugita. Ecology 71:1204–1207
West GB, Brown JH, Enquist BJ (1997) A general model for the origin of allometric scaling laws in biology. Science 276:122–126
West GB, Brown JH, Enquist BJ (1999) A general model for the structure and allometry of plant vascular systems. Nature 400: 664–667
White J (1981) The allometric interpretation of the self-thinning rule. J Theor Biol 89:475–500
Whitfield J (2001) All creatures great and small. Nature 413:342–344
Xue L, Hagihara A (2002) Growth analysis on the C-D effect in self-thinning Masson pine (Pinus massoniana) stands. For Ecol Manage 165:249–256
Yang Y, Titus StJ (2002) Maximum size–density relationship for constraining individual tree mortality functions. For Ecol Manage 168:259–273
Yoda KT, Kira T, Ogawa H, Hozumi K (1963) Self-thinning in overcrowded pure stands under cultivated and natural conditions. J Inst Polytech (Osaka University) D 14:107–129
Zeide B (1985) Tolerance and self-tolerance of trees. For Ecol Manage 13:149–166
Zeide B (1987) Analysis of the 3/2 power law of self-thinning. For Sci 33:517–537
Zeide B (2001) Natural thinning and environmental change: an ecological process model. For Ecol Manage 154:165–177
Zeide B (2004) How to measure stand density. Trees (in press)
Acknowledgements
The author wishes to thank the Deutsche Forschungsgemeinschaft for providing funds for forest growth and yield research as part of the Sonderforschungsbereich 607 “Growth and Parasite Defense” and the Bavarian State Ministry for Agriculture and Forestry for permanent support of the Forest Yield Science Project W 07. Prof. Dr. Hermann Spellmann of the Lower Saxony Forest Research Station in Göttingen complemented the Bavarian dataset with two experimental plots from the former Prussian Forest Research Station. Thanks are also due to Prof. Dr. Boris Zeide for helpful discussion, Hans Herling for preparation of graphs and anonymous reviewers, for constructive criticism.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Christian Koerner
Communicated by Christian Koerner
Electronic Supplementary Material
Rights and permissions
About this article
Cite this article
Pretzsch, H. Species-specific allometric scaling under self-thinning: evidence from long-term plots in forest stands. Oecologia 146, 572–583 (2006). https://doi.org/10.1007/s00442-005-0126-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00442-005-0126-0