Introduction: Stand density management is essential for adaptive silviculture, thinning decisions, growth modeling, and yield prediction in forestry, particularly for plantations. Despite extensive research on self-thinning rules and the maximum size-density law, significant gaps remain in the biophysical understanding and validation of the relationships among key stand variables and parameters.Methods: This study theoretically explored and validated the relationship between maximum size-density and two key metrics: average diameter at breast height (D) and tree height (H). We used time-series data from a 30-year clear-cut, fully stocked Chinese fir plantation, a fast-growing commercial species in China, for validation.Results: A growth balance status for fully stocked stands was proposed, wherein prior to self-thinning, the growth rate of the stand basal area (G) aligns with that of the average tree height (H), expressed as G'/(G−b0)=H'/H and approaching a constant slope, b₁. Generalized maximum size-density and stand density index (SDI) equations were developed: N1.0=A×D−2 and SDI=A⋅D−2 with A=4×(b0+b1H)/π, differing from traditional equations. Additionally, a generalized self-thinning equation, v=kHqN1.0−1 or w=c1HqN1.0−1, was introduced, indicating that in fully stocked stands, tree volume or biomass depends on both tree height and tree count.Discussion: These findings advance understanding of the maximum size-density law and self-thinning boundary, providing refined tools for managing stand density in Chinese fir plantations.

A suitable self‐thinning model is fundamental to effective density control and management. Using data from 265 plot measurements in oak mixed forests in central China, we demonstrated how to estimate a suitable self‐thinning line for oak mixed forests from three aspects, i.e., self‐thinning models (Reineke's model and the variable density model), statistical methods (quantile regression and stochastic frontier analysis), and the variables affecting stands (topography and stand structure factors). The proposed variable density model, which is based on the quadratic mean diameter and dominant height, exhibited a better goodness of fit and biological relevance than Reineke's model for modeling the self‐thinning line for mixed oak forests. In addition, the normal‐truncated normal stochastic frontier model was superior to quantile regression for modeling the self‐thinning line. The altitude, Simpson index, and dominant height–diameter ratio (/D) also had significant effects on the density of mixed forests. Overall, a variable density self‐thinning model may be constructed using stochastic frontier analysis for oak mixed forests while considering the effects of site quality and stand structure on density. The findings may contribute to a more accurate density management map for mixed forests.

The self‐thinning rule in forest stands is fundamental to the development of density management strategies, as it determines the maximum stand density achievable for a given tree size. Accurate modeling of the maximum density line is crucial, but selecting representative data points for this purpose remains a challenge. Using 18 years of data from five Cunninghamia lanceolata plantations with varying initial planting densities, this study investigated whether relationships between mean tree basal area (g) and height (H) can reveal forest developmental stages and identify when stands begin self‐thinning and reach maximum density. Our results showed a significant linear relationship (p < 0.05) between g and H after self‐thinning was established, supporting the presence of self‐regulatory growth mechanisms. These findings enabled the development of a novel sample selection method for constructing more accurate maximum density line models, outperforming traditional methods that rely on arbitrary thresholds. Additionally, we derived formulas to describe total stand basal area (G_1.0) during different growth stages, revealing positive correlations with mean height during early growth and negative correlations with mean diameter during self‐thinning. This research advances the understanding of self‐thinning dynamics and provides practical tools for improving density management in plantation forestry.

Stand density management is important for decision-making regarding adaptive silviculture and thinning, growth modelling, and yield prediction in forests, especially plantations. Although substantial research related to the self-thinning rule and maximum size-density law has been conducted, there are still critical gaps that exist in the biophysical explanation and validation of the relationships among stand variables and relevant parameters. In this study, time series observations from six plots of fully stocked Chinese fir plantations with different densities of planted trees were used to characterise the growth of stand basal area (G), average height (H), and diameter at breast height (D). The growth trends in the stand parameters and the relationships among them were analysed. As indicated by previous studies, in the fully stocked stands, there was a significant linear relationship between G and H. This study also resulted in the following new findings: (1) At the beginning, the growth rate of stand basal area ($P_G$) was greater than the growth rate of average height ($P_H$), but $P_G$ decreased quickly as the stands approached canopy closure and then became stable. Meanwhile, as the stands neared canopy closure, the rate of increase in the G/H ratio decelerated, ultimately resulting in a stable G/H value that approached the first limit value. This led to a stand growth balance status that continued until self-thinning took place. (2) Artificial thinning broke the growth balance status, but the stands returned to balance status if they were still young enough. Self-thinning also broke the growth balance status and lead to fluctuating growth rates of both G and H, but the fluctuations were very slight, which showed a trend in similar growth rates of G and H. (3) The findings implied that the stand G and H growths were allometric at the beginning but became isogonic as canopy closure and self-thinning were approached. On the other hand, the H growth rate was generally greater than that of D, but both growth rates showed a trend in similar values after the stands matured. Subsequently, the H/D ratio is anticipated to stabilize and gradually converge towards the second limit value once the stands reach maturity. The results implied that the stand growth balance status and two limit values can be used to identify and select fully stocked stands that are needed for the development of maximum size-density equations and self-thinning rules.