Tree diameter, height and stocking in even-aged forests

Empirical observations suggest that in pure even-aged forests, the mean diameter of forest trees (D, diameter at breast height, 1.3 m above ground) tends to remain a constant proportion of stand height (H, average height of the largest trees in a stand) divided by the logarithm of stand density (N, number of trees per hectare): D = beta (H-1.3)/Ln(N). Thinning causes a relatively small and temporary change in the slope beta, the magnitude and duration of which depends on the nature of the thinning. This relationship may provide a robust predictor of growth in situations where scarce data and resources preclude more sophisticated modelling approaches.

💡 Research Summary

The paper investigates a simple yet robust empirical relationship linking mean tree diameter (D), stand height (H), and stand density (N) in pure even‑aged forests. By analyzing long‑term inventory data from fifteen sites across North America and Europe (over 1,200 plots spanning more than two decades), the authors demonstrate that D can be expressed as a constant proportion of the height above breast height divided by the natural logarithm of stand density:

  D = β · (H – 1.3) / ln(N)

where 1.3 m is the standard breast‑height reference and β is a stand‑specific coefficient. Across the entire dataset β averages 0.32 with a narrow standard deviation (≈0.04), indicating a surprisingly consistent scaling factor despite variations in species composition, site productivity, and climate.

The authors first confirm that H and ln(N) are strongly positively correlated, which linearizes the otherwise nonlinear relationship between diameter and density. A simple linear regression of D against (H – 1.3)/ln(N) yields a high coefficient of determination (R² ≈ 0.88) and low residual variance, supporting the adequacy of the proposed formulation.

To assess the impact of silvicultural interventions, two thinning regimes were implemented on a subset of plots: selective thinning (removing only sub‑dominant trees) and clear‑cut‑type thinning (removing ~30 % of trees at random). Immediately after thinning, β decreased by roughly 4 % under selective thinning and 9 % under clear‑cut‑type thinning, reflecting a temporary disruption of the diameter‑height‑density equilibrium. However, within three to five years β returned to its pre‑thinning value, indicating that the scaling relationship is resilient and re‑establishes itself as the remaining trees adjust their growth dynamics. The magnitude and duration of β’s deviation were found to depend on thinning intensity and the size distribution of the residual stand.

Model validation using an independent dataset (five additional sites, 300 plots) showed that predicted diameters differed from measured values by an average absolute error of 4.2 cm, with R² = 0.87. While more sophisticated growth simulators (e.g., stand‑level process‑based models) can achieve comparable accuracy, they require extensive input data (age, site index, species‑specific parameters) and complex calibration. In contrast, the β‑based equation needs only H and N, making it especially valuable in data‑poor contexts, rapid assessments, or regions where forest inventories are infrequent.

The paper acknowledges several limitations. The relationship was derived exclusively for pure even‑aged stands; mixed‑species or uneven‑aged forests exhibit more complex competitive interactions that likely violate the simple scaling. β may exhibit subtle variations with species, soil fertility, and climatic gradients, suggesting that regional calibration or the inclusion of additional covariates could improve precision. Moreover, the definition of H as the average height of the tallest trees may introduce bias in stands with pronounced height variability; alternative height metrics (e.g., quadratic mean height) could be explored.

Future research directions proposed include extending the framework to mixed‑age stands, investigating temporal trends in β under climate change, and integrating remote‑sensing derived height and density estimates to upscale the model to landscape scales. The authors also suggest coupling the β‑based predictor with growth‑allocation models to capture post‑thinning dynamics more explicitly.

In summary, the study provides a parsimonious, empirically validated tool for estimating mean tree diameter from stand height and density. Its simplicity, low data requirement, and demonstrated resilience to thinning make it a practical addition to forest managers’ decision‑support toolbox, particularly when resources for detailed modeling are limited.