A Unified Law of Spatial Allometry for Woody and Herbaceous Plants

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Abstract
The objective of the present paper is to provide both proof and theoretical deduction of an overlapping, valid law of allometry for woody and herbaceous plants used in agriculture and forestry. In his attempt to find an adequate expression for stand density, independent of site quality and age, Reineke (1933[281]) developed the following equation for even‐aged and fully stocked forest stands in the northwest of the USA: ln(N) = a ‐ 1.605 . ln(dg), based on the relationship between the average diameter dg and the number N of trees per unit area. With no knowledge of these results, Kira et al. (1953[281]) and Yoda et al. (1957[281] and 1963[281]) found the boundary line ln(m) = b ‐ 3/2 . ln(N) in their study of herbaceous plants. This self‐thinning rule ‐ also called the ‐ 3/2‐power rule ‐ describes the relationship between the average weight m of a plant and the density N in even‐aged herbaceous plant populations growing under natural development conditions. It is possible to make a transition from Yoda's rule to Reineke's stand density rule if mass m in the former rule is substituted by the diameter dg. From biomass analyses for the tree species spruce (Picea abies [L.] Karst.) and beech (Fagus sylvatica L.), allometric relationships between biomass m and diameter d are derived. Using the latter in the equation ln(m) = b ‐ 3/2 . ln(N) leads to allometric coefficients for spruce (Picea abies [L.] Karst.) and beech (Fagus sylvatica L.), that come very close to the Reineke coefficient. Thus Reineke's rule (1933[281]) proves to be a special case of Yoda's rule. Both rules are based on the simple allometric law governing the volume of a sphere v and its surface of projection s: v = c1 . s3/2. If the surface of projection s, is substituted by the reciprocal value of the number of stems s = 1/N and the isometric relationship between volume v and biomass m is considered v = c2 . m1.0 we come to Yoda's rule m = c3 . N‐3/2 or, in logarithmic terms, ln(m) = ln c3 ‐ 3/2 . ln(N).