, Volume 146, Issue 4, pp 572–583 | Cite as

Species-specific allometric scaling under self-thinning: evidence from long-term plots in forest stands

  • Hans PretzschEmail author
Ecosystem Ecology


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.


Allometry Self-thinning Stand density rule −3/2-Power law Euclidian geometrical scaling Fractal scaling 



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.

Supplementary material

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Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  1. 1.Chair of Forest Yield Science, Faculty of Forest Science and Resource ManagementTechnical University of MunichFreisingGermany

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