We tested whether the self-thinning relationship varied among tree species. Because nonlinearity in this relationship may cause apparent differences in the allometric coefficient estimates reported for a given species, we also tested the linearity of this relationship. We used a homogeneous dataset based on NFI data for 11 temperate and Mediterranean species and estimated self-thinning relationships using stochastic frontier analysis (SFA), a relevant technique for assessing conditional laws of the maximum.
Suitability of NFI data to estimate self-thinning relationships
NFI data have never been used to fit self-thinning relationships. For all species, realistic raw lnN–lnD
g relationships were found (Fig. 1). Concern for self-thinning conditions to be found along the entire range of diameters considered led us to set upper/lower thresholds on the quadratic mean diameter based on two criteria: (1) avoid plots where stand density could be underestimated (NFI lower countable threshold) and where it may result in an artificial curvature of the lnN–lnD
g relationship at low diameters (Schütz and Zingg 2010) and (2) observe true mortality in the densest 10% of plots for each lnD
g class. The observed mortality rates (2.45% on average over the last 5 years) indicated that self-thinning conditions were likely, even if they constituted no formal evidence for it. Our aim was only to discard diameter classes where self-thinning was not likely to occur and not to select stands experiencing self-thinning. The diameter ranges selected were sometimes very restricted (e.g. sessile oak) because no mortality could be observed in large diameter classes, which is a limit of NFI data.
Control of site conditions
The intercept of the self-thinning relationship has been shown to depend on site fertility (Bi 2004; Weiskittel et al. 2009). The aim of our study was not to investigate variations in the self-thinning relationship with site conditions but to estimate and compare the maximum species boundaries (sensu Weller 1990) over their distributional range in France. However, imbalances in site conditions over the diameter range may induce biases or artificial curvature in the self-thinning relationship estimates. We controlled for these imbalances using the French biophysical classification of forests (Inventaire Forestier National 2009; see Section 2).
Relative position of linear self-thinning relationships
The maximum attainable density is known to be positively related to species shade tolerance (Jack and Long 1996). The species ranking at D
g = 20 cm showed a significant positive correlation (r = 0.62, p = 0.04) with the degree of shade tolerance (Niinemets and Valladares, 2006; http://www.esapubs.org/archive/mono/M076/020/appendix-A.htm). As shade-tolerant species, Norway spruce, silver fir and common beech displayed self-thinning relationships that occupied a higher position in the lnN–lnD
g plane (Fig. 2) than those of light-demanding pine and oak species. However, Douglas fir, Scots pine and Corsican pine reached unexpectedly high maximum densities given their low shade tolerance, suggesting that factors other than light resource may favour stocking (Shirley 1943; Zeide 1985). The specific differences in crown allometry and growing space efficiency might also be a promising perspective in the understanding of specific differences in the self-thinning relationship (Pretzsch and Schutze 2005). Indeed, recent works have pointed out that these specific characteristics affected the packing capacity of a stand. Pretzsch and Schutze (2005) observed that common beech had higher space occupancy than Norway spruce and was less efficient in space exploitation due to differences in branching and resource allocation. This was shown to affect the maximum stocking (Schütz and Zingg 2010).
Species variations in allometric coefficient of self-thinning
The significant between-species differences in the self-thinning allometric coefficient invalidate the hypothesis of a constant coefficient among species. The species classifications with respect to the allometric coefficient estimates were similar for 3 species to the classifications of Pretzsch and Biber (2005) who used long-term experimental data from fully stocked stands: common beech < Norway spruce < Scots pine. This remarkable agreement confirms the reliability of NFI data for such assessment and indicates some proximity between the self-thinning boundary sensu Weller (1990) and the dynamic self-thinning trajectory of a given species. The slope estimate we found for sessile oak was not the flattest as in Pretzsch and Biber (2005), but this disagreement might be due to the short diameter range covered by our dataset for this species. Groups of non-distinguishable species were identified (up to five species, Table 4), highlighting the necessity of a large number of species to avoid selection biases and properly address the issue. As a main outcome of this study, the existence of non-overlapping groups definitely confirms significant differences in the self-thinning relationship between species.
Despite the common classification with Pretzsch and Biber (2005) for three species, our slope estimates were all steeper by 1–12.8%. A formal comparison between their dynamic self-thinning trajectories and our maximum self-thinning boundaries is difficult. However, the minimum mean diameters in their study were below 15 cm for the four species (from 5.7 to 13.4 cm). The curvilinearity of the lnN–lnD
g relationship evidenced in this study may explain the difference as it would generate flatter slope estimates with lower diameter thresholds and a linear assumption. This might, however, not be the only explanation as the slope we found for sessile oak was steeper, albeit without curvature.
Stand origin was shown to affect the slope of the self-thinning line for Douglas fir, with planted stands having a flatter slope (Weiskittel et al. 2009). In France, Douglas fir is only found in plantations, which may result in an artificially lower slope than for the other species. Corsican and maritime pines are also mostly planted, but a few naturally regenerated stands may be found in the resource as we observed dense young stands (Fig. 1). Their slope estimates might thus not be biased.
Curvilinearity of the self-thinning relationship
For 7 out of 11 species, the self-thinning relationship was better represented by a quadratic relationship of lnD
g than a linear one. This highlighted the effect of developmental stage on the self-thinning rate. Curvilinearity of the self-thinning relationship has previously been evidenced for common beech, Norway spruce (Pretzsch 2006; Schütz and Zingg 2010) and Scots pine (Pretzsch 2006). Interestingly, we confirmed the allometric relationship found for sessile oak (Pretzsch 2006). Zeide (2010) suggested an interpretation for the increasing self-thinning rate with stand ageing: (1) as trees get bigger, the gaps created by dead trees widen and (2) in parallel, their higher metabolic cost of maintenance and reduced photosynthetic activity (Yoder et al. 1994) would make bigger trees less able to react to canopy openings. Consequently, an opening in old stands would lead to overstocked yet relatively sparser stands. Although Nilson’s (1973) model has been shown to be particularly appropriate to account for the continuous increase in the self-thinning rate with developmental stage (Zeide 2010), it could not be tested with the SFA method because of its nonlinearity in the parameters. The quadratic term in lnD
g used here also resulted in a curvature of the self-thinning relationship with tree diameter.
Comparison of species self-tolerance over developmental stage
We found that self-tolerance decreases with developmental stage for several species. The decrease varied from very strong (Scots pine, Douglas fir and silver fir) to null (Corsican pine, pedunculate, pubescent and sessile oaks; Fig. 4), leading to a dramatic change in the ranking of species self-tolerance over developmental stage. This suggests that absolute species comparisons are not meaningful.
Variations in species maximum density with developmental stage have implications for the dynamics of mixed-species communities whose stability or evolution towards pure communities may depend on the relative maximal densities that the admixed species can each tolerate (Shaw 2006). The significant differences in species self-thinning relationships and in their ordering over ontogeny (Figs. 3 and 4) suggest that the dynamics of corresponding mixtures may vary over time. Given that species also differ in their intrinsic growth rate (Lambers and Poorter 1992), a comprehensive assessment of the dynamical consequences of self-thinning variations among tree species, including laws of diameter growth, remains to be conducted.