, Volume 26, Issue 2, pp 469–478

Models relating stem growth to crown length dynamics: application to loblolly pine and Norway spruce


    • US Forest Service
  • Annikki Mäkelä
    • Department of Forest EcologyUniversity of Helsinki
  • Edwin J. Green
    • Department of Ecology, Evolution and Natural Resources, Cook College Rutgers University
  • Ralph L. Amateis
    • Department of Forest Resources and Environmental ConservationCollege of Natural Resources and Environment
  • Harri Mäkinen
    • Vantaa Research Unit PL 18The Finnish Forest Research Institute
  • Mark J. Ducey
    • Department of Natural Resources and the EnvironmentUniversity of New Hampshire
Original Paper

DOI: 10.1007/s00468-011-0608-0

Cite this article as:
Valentine, H.T., Mäkelä, A., Green, E.J. et al. Trees (2012) 26: 469. doi:10.1007/s00468-011-0608-0


We derive and analyze a model that relates the growth rate of cross-sectional area (‘csa’) at any height on the central stem of a tree to crown-length dynamics. The derivation is based, in part, on assumptions that (a) active csa on the central stem relates allometrically to the length of crown above the cross section, and (b) inactive csa is proportional to active csa within the crown. We also assume that the deactivation rate of csa beneath the crown is determined, in part, by the rate of crown rise. Integration of the growth-rate model under an additional assumption—that total crown length is constant after stand closure—provides a simple model of annual or periodic growth of total csa that can be fit to standard growth data. Three implications of the assumptions and integration are notable: (1) total csa within the crown scales allometrically with stem length above the cross section; (2) for a special case, total csa beneath the crown scales with stem length above the cross section; more generally, csa scales with a linear combination of the stem and crown lengths; and (3) the stem beneath the crown forms to approximate a frustum of a quadratic paraboloid. Basal area data from a loblolly pine (Pinus taeda L.) spacing trial show good agreement with (1) and (2), and with an empirical model developed from the special case of (2). Data from the plots of a Norway spruce (Picea abies (L.) Karst.) thinning trial, where crown length remained approximately constant, show good agreement with (2) and the empirical model. Prediction (3) is demonstrated by simulation.


AllometryGrowth modelPipe modelScalingStem form

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© Springer-Verlag (outside the USA) 2011