The Allometric Quarter-Power Scaling Model and Its Applicability to Grand Fir and Eucalyptus Trees

  • Hannah Capes
  • Robert J. Maillardet
  • Thomas G. Baker
  • Christopher J. Weston
  • Don McGuire
  • Ian C. Dumbrell
  • Andrew P. Robinson


The use of the allometric model \(y = \beta x ^{\alpha }\) to describe the relative growth of morphological traits of trees is a source of contention in ecology. This is particularly so in a specific form, the West, Brown and Enquist model, which predicts values of \(\alpha \) that are multiples of 1/4 for various allometric relationships—the quarter-power scaling law. We use statistical techniques to test the appropriateness of the quarter-power scaling allometric model in a number of different relative growth relationships of trees. Two separate datasets are used, one of repeated measures of Abies grandis (Grand fir) trees, another of independent measures of Eucalyptus trees. Nonlinear mixed-effects modelling is used to fit allometric models to the datasets. Generalised additive models, equivalence testing and traditional significance testing are used to assess the adequacy of the allometric models fitted and the values of the estimated exponents relative to those predicted by the WBE model. In only one of the five models fitted was there empirical evidence for the WBE-predicted quarter-power exponent. However, the adequacy of the allometric models was generally supported, though a need for further analysis over a larger range of tree ages/sizes is indicated.

Supplementary materials accompanying this paper appear online.


Allometric scaling Quarter-power scaling law Nonlinear mixed-effects models Equivalence testing 



The authors are grateful to S. Livesley (the University of Melbourne) and S. Elms (HVP Plantations Pty Ltd) for contributing portions of the analysed datasets. The clarity of the paper has been greatly improved by the thoughtful comments of two anonymous reviewers.

Supplementary material

13253_2017_292_MOESM1_ESM.pdf (1.4 mb)
Supplementary material 1 (pdf 1398 KB)


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Copyright information

© International Biometric Society 2017

Authors and Affiliations

  • Hannah Capes
    • 1
  • Robert J. Maillardet
    • 2
  • Thomas G. Baker
    • 3
  • Christopher J. Weston
    • 2
  • Don McGuire
    • 4
  • Ian C. Dumbrell
    • 5
  • Andrew P. Robinson
    • 6
  1. 1. Empirica ResearchMelbourneAustralia
  2. 2. University of Melbourne (UOM)ParkvillleAustralia
  3. 3. UOMRichmondAustralia
  4. 4. Synergetic Forestry Services Mount GambierAustralia
  5. 5. Forest Products CommissionKensingtonAustralia
  6. 6. CEBRA, School of BioSciencesUOMParkvillleAustralia

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