One Forest, Three Tree Species, Three Very Different Growth Equations

Measuring how tall a tree stands is one of the most consequential challenges in tropical forestry. A tape or laser can reveal a trunk’s diameter quickly, but obtaining the full height of a forest giant usually requires climbing, specialized equipment, or laborious field procedures. That imbalance has made mathematical models essential for translating easy-to-measure tree diameters into estimates of stature, forest structure, biomass and carbon stocks. A new study in Kenya’s Kakamega Forest shows why those models cannot safely be treated as universal: three dominant tree species displayed markedly different relationships between diameter and height, and the best-performing equations for one species worked poorly for another.

Researchers examined Pterocarya stenoptera, Spathodea campanulata and Antiaris toxicaria in a one-hectare permanent plot in the western Kenyan forest. They measured diameter at breast height, or DBH, for trees with trunks at least 10 centimetres across and used a laser hypsometer to determine total height. The study then fitted four candidate height–diameter models: two nonlinear forms, Weibull and Chapman–Richards, and two forms that can be expressed linearly, Power and Logarithmic. This broad comparison allowed the investigators to test not merely which equation looked best in a statistical sense, but which one would remain useful when predicting heights for trees it had not encountered during model fitting.

Model quality differed sharply among the species. For Spathodea campanulata, the Power equation produced the strongest predictive performance, with an adjusted coefficient of determination, or adjusted R², of 0.698. That means the model accounted for a substantial share of the observed variation in tree height within the sampled trees. The same functional form gave the best reported result for Antiaris toxicaria, although its adjusted R² was much lower, at 0.366. In other words, diameter was a more informative predictor of height for the former species than for the latter, despite the common model structure. A model can therefore appear effective for one tree while still providing only moderate information for another growing in the same forest.

The most striking result came from Pterocarya stenoptera. Every candidate model produced a negative adjusted R² for this species, indicating that none predicted its height better than a simple benchmark based on the sample mean. That does not mean diameter is unrelated to height in the species, but it does mean the available observations did not support any of the tested equations as reliable height predictors. Cross-validation confirmed substantial differences among species in predictive accuracy, while bootstrap resampling revealed greater uncertainty in the fitted parameters for P. stenoptera. The researchers linked that instability to the species’ smaller sample size and more variable height–diameter relationship.

These contrasts expose a hidden weakness in broad-scale forest inventories. One common allometric equation may be applied to many taxa for convenience, especially when time, equipment and trained personnel are limited. Yet the study indicates that generalized equations can miss important local differences in growth architecture. In Kakamega Forest, the same mathematical form could describe two species with markedly different explanatory power, and a third species required caution against prediction altogether. Species identity, local growth conditions and the range of tree sizes represented in a dataset all need to be considered before an equation is transferred from a research plot to inventory work.

The study’s technical evaluation addressed several threats to the reliability of the models. The researchers compared fits using adjusted R², root mean square error and Akaike Information Criterion, and tested performance with leave-one-out cross-validation, in which a model is repeatedly evaluated after excluding one observation at a time. Bootstrap resampling was used to examine parameter uncertainty. Residual heteroskedasticity—the tendency for prediction errors to change systematically across the range of diameter—was assessed with the Breusch–Pagan test and corrected where necessary. These procedures helped distinguish a superficially attractive equation from one that is stable and informative for the population being sampled.

Even the successful models should not be mistaken for complete descriptions of forest architecture. An adjusted R² of 0.698 for S. campanulata is a meaningful result, but it also leaves much of the observed variation unexplained. Heights can differ among trees of the same diameter because of differences in site conditions, crown position, past disturbance, competition, genetics and other factors that a model based only on DBH cannot capture directly. The lower explanatory power for A. toxicaria and the failure of the tested equations for P. stenoptera further caution against interpreting a fitted curve as a biological constant. Each equation is best understood as a locally calibrated statistical summary of a particular population.

The practical value of the work lies in creating tools that reflect what the Kakamega data actually show. Species-specific models can improve estimates of canopy structure in the studied forest and provide a foundation for future biomass estimation, where trunk dimensions and height are combined to estimate the amount of vegetation and carbon stored in trees. Better forest structure estimates may also support repeated monitoring, allowing managers to track changes in dominant trees and evaluate whether conservation measures are working. The researchers note that independent validation will be needed before the models can support wider applications, particularly in sampled populations, other locations or changing environmental conditions.

That local focus is also the study’s central conservation message. Kakamega Forest is home to globally important tropical biodiversity, and the authors connect the research to Sustainable Development Goal 13 on climate action and Goal 15 on life on land. Precise structural measurements can improve ecosystem management by giving researchers and forest authorities a more faithful picture of forest stature and biomass. The work does not establish a new rule for estimating trees, nor does it imply that one species can serve as a proxy for an entire forest. Instead, it demonstrates that effective monitoring requires models tailored to the biological differences among dominant species, followed by rigorous testing beyond the plot in which they were developed.

Forests are often described through aggregate numbers—tree counts, timber volume, canopy cover and stored carbon—but each number depends on assumptions about how trees grow. The Kakamega results make those assumptions visible. S. campanulata displayed a comparatively strong and predictable relationship between diameter and height; A. toxicaria had a weaker relationship, though the researchers reported comparatively lower absolute prediction error; and P. stenoptera showed a highly variable pattern that none of the tested models could describe reliably. The result is a scientific reminder that a forest is not architecturally uniform. Its dominant species may not rise at the same rate, respond similarly to competition or fit the same statistical curve. For a warming planet under growing pressure, capturing those differences is not a technical footnote: it is a condition of knowing what the forest actually contains, where its carbon is concentrated and how faithfully that stock can be monitored over time.

Subject of Research: Species-specific tree height–diameter relationships in Kakamega Forest, Kenya

Article Title: Species specific height diameter relationships of dominant tropical rainforest trees in Kakamega Forest Kenya

Article References: Nyongesa, J. M., Oronyi, W., Oyoko, E., Ronoh, E. K., Mwalati, L., Vincent, S., Gayo, L., Li, Y., & Otieno, D. O. (2026). Species specific height diameter relationships of dominant tropical rainforest trees in Kakamega Forest Kenya. Discover Sustainability. https://doi.org/10.1007/s43621-026-04832-z

Image Credits: AI Generated

DOI: 10.1007/s43621-026-04832-z

Keywords: Kakamega Forest, tree allometry, height–diameter models, tropical rainforest, forest inventory, biomass estimation, forest monitoring, Pterocarya stenoptera, Spathodea campanulata, Antiaris toxicaria, Species, specific