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The self-thinning law, which is also referred to as the 3/2 power law of self-thinning (YODA et al., 1963), describes an increase in mean plant weight (w) with decreasing plant density (Nu) in unthinned (overcrowded) pure stands: w = kNu (1) where k and α are constants that represent the self-thinning coefficient and the self-thinning exponent, respectively. The name of the law derives from the fact that α = 3/2 for many plants. A comparable law has been applied to the relationship between other measurements of mean size and plant density. REINEKE (1933) described the decrease in Nu with increasing quadratic mean diameter (Dq, which equals the diameter at breast height for a tree with the mean basal area) in overcrowded pure stands as follows: lnNu = β-γlnDq (2) where β and γ are constants. REINEKE (1933) considered the exponent γ to be universal constant (i.e. γ is 1.605). Many researchers have debated the mechanisms responsible for generating the self-thinning exponents α and γ (e.g., SATOO, 1962; WELLER, 1987; OSAWA and ALLEN, 1993; ENQUIST et al., 1998). These studies have adopted three main approaches: an early isometric approach (YODA et al., 1963), an allometric approach (SATOO, 1962; WELLER, 1987; OSAWA and ALLEN, 1993), and more recently a fractal approach (ENQUIST et al., 1998). It should be noted that all three approaches share the assumption that total stand leaf mass remains constant during stand development. These studies also assumed that the power relationships in equations (1) and (2) are valid, although the values of α and γ vary slightly rather than being universally valid constants. However, several studies have questioned the validity of the power relationship for the Dq-Nu relationship (e.g., MEYER, 1938; LAASASENAHO and KOIVUNIEMI, 1990; ZEIDE, 1995, 2005, 2010; del RÍO et al., 2001; CHARRU et al., 2012). These studies have reported nonlinearity of the DqNu relationship on a log-log scale even though equation (2) indicates that the relationship should be linear on a log-log scale. The nonlinearity may be due to temporal variability in crown closure and stand leaf mass (ZEIDE, 1995, 2005; del RÍO et al., 2001). This can be understood if we consider three phases of stand development that are related to a stand’s self-thinning and crown closure dynamics (del RÍO et al., 2001; VANDER SCHAAF and BURKHART, 2008) (Fig. 1): (1) From the young stage until crown closure (Phase B in Fig. 1), self-thinning is less than the mortality rate that occurs at the maximum competition for light between individual Corresponding author: Tomohiro Nishizono * Department of Forest Management, Forestry and Forest Products Research Institute, Tsukuba, Ibaraki 305–8687, Japan E-mail: nishizo@ffpri.affrc.go.jp * Kansai Research Center, Forestr y and Forest Products Research Institute, Kyoto 612–0855, Japan ABSTRACT

journal of forest planning

Does the Relationship between Quadratic Mean Diameter and Stem Density in Old Thinned and Unthinned Cryptomeria japonica Forests Deviate from a Power Function?