This study was conducted in the Longuza and Mtibwa forest plantations, both managed by the Tanzania Forest Services Agency (TFS).

Longuza, located on the eastern (windward) slopes of the Usambara Mountains along the coast, receives high precipitation that strongly favours teak growth. In contrast, Mtibwa is situated inland, far from the coast, and experiences relatively lower rainfall. Both sites are positioned on the windward sides of mountainous landscapes: Mtibwa lies east of the Mkingu Nature Reserve, while Longuza is east of the Amani Nature Reserve.

Mtibwa Forest Plantation is located inland, east of Mkingu Nature Reserve, relatively far from the coast in the Morogoro Region, on the eastern slopes of the Nguu Mountains within Mvomero District. The plantation lies between latitudes 6˚00’ and 6˚10’S and longitudes 37˚40’ and 37˚45’E. Teak was first introduced at Mtibwa in 1936 through an 8-acre trial plot, followed by the establishment of additional experimental plots in 1954, 1955, and 1957 at the Lusunguru block. Large-scale teak planting commenced in 1961 at the Mtibwa block. The plantation covers approximately 16,065.6 ha and, for management purposes, is subdivided into three blocks: Mtibwa (1023.6 ha), Lusunguru (2092 ha), and Pagale (12,950 ha). Teak is the dominant species, accounting for about 95% of the planted area. The area receives long rains from March to May, followed by a long dry season lasting until October. Short rains are from November to December. The average mean annual rainfall is 1217 mm. Since this is marginal for Teak growth, it is supplemented by sub-surface underground water. Temperatures are high, particularly from January to March. Mid-day temperatures vary from 14˚C to 36˚C. The soils in Mtibwa are sandy entisols and vertisols whereas in water-logged plains, inceptisol loams are more predominant [15].

Longuza Forest Plantation is located in Muheza District, Tanga Region, northeastern Tanzania, between latitudes 4˚55’ and 5˚10’S and longitudes 38˚40’ and 39˚00’E, at elevations ranging from 160 to 560 m above sea level. The plantation is situated at the foothills of the eastern Usambara Mountains and is influenced by a coastal windward climatic regime, which contrasts markedly with the climatic conditions prevailing at Mtibwa. Similar to Mtibwa, teak is the principal tree species in Longuza, occupying approximately 96% of the planted area. The mean maximum temperature in the area ranges from 26˚C to 32˚C, while the mean minimum temperature ranges from 15˚C to 20˚C. The mean annual rainfall is 1548 mm with a dry spell between June and September. The area experiences short rain from October to December and long rain from March to May. The soil type in the study area are predominantly sandy clay loam. They exhibit dark reddish-brown to deep red hues, with coloration intensifying at greater depths. Soil depth is highly variable, spanning from shallow profiles of less than 20 cm to very deep profiles exceeding 120 cm [16]. The study site is shown in Figure 1.

Figure 1.Study sites.

A systematic sampling design was applied within each plantation compartment. In this study, a compartment refers to a forest plantation management unit composed of a single tree species of uniform age. This study was conducted in compartments containing stands that are at least 5 years old to generate forest inventory data for the preparation of forest management plans for Mtibwa and Longuza forest plantations. Sampling intensity was age-dependent: a 2% intensity was applied to compartments aged 5 - 12 years, while a 5% intensity was applied to compartments aged 12 years or older. Plot density varied among compartments according to compartment area and the selected sampling grid, with inter-plot spacing of 150 m × 100 m for compartments aged 5 - 12 years and 100 m × 60 m for compartments older than 12 years. All plots were circular with a radius of 9.8 m. In total, 97 teak-planted compartments were inventoried, comprising 2462 sample plots.

Within each sample plot, site and stand attributes were recorded, including geographic coordinates, slope, stand age, planted species, and management interventions such as pruning and thinning. Diameter at breast height (D) was measured for all trees within the plot. Tree height (H) was measured for a subset of five trees per plot. Three of these trees were selected from the largest-diameter individuals to estimate dominant height (HD), consistent with the standard practice of sampling 100 dominant trees per hectare, given the plot size of 0.03 ha [17]. The remaining two trees were selected to represent intermediate and suppressed size classes. Trees exhibiting broken tops or stem deformities were excluded from height measurements.

With site classification based on stand age and the mean height of dominant (DH) trees measured at the plot level. For each plot, the observed age and mean HD were compared against site class curves derived from the yield table developed for Longuza and Mtibwa plantation [18]. The yield table defines three site classes corresponding to reference H of 18, 22 and 26 m. Each plot was assigned to the site class that minimised the absolute difference between the observed mean DH and the DH predicted by Equation (1).

where the RH is the reference tree height, DH is the predicted dominant height, and A is age.

Once the site class of each plot was determined, three trees measured for H were selected for H-D development. This included one sample tree randomly chosen from the three largest trees, and two sample trees that did not participate in determining the site class, i.e., medium, and the smallest. Table 1 presents the descriptive statistics summary of sample trees for H-D development for each site class. The scatter plot of the D versus H of each site class is presented in Figure 2. In addition, a sensitivity analysis was conducted to assess the reliability of the site class allocation. This involved randomly adding or subtracting one meter to the dominant height values, assuming this to represent potential measurement error. The results showed that out of 2462 plots, only 56 plots (2.27%) were reassigned to a different site class. A realistic measurement error of ±1 m only affected a small proportion of plots, meaning that minor field measurement inaccuracies are unlikely to substantially affect site productivity classification.

Table 1.Descriptive statistics summary of sample trees.

n: number of observations; SD: Standard Deviation.

Figure 2.Scatter plot of height versus diameter.

2.5.1. Fitting the H-D Models

Model development was carried out in two phases. In the first phase, a general H-D model was fitted using the pooled dataset from all site classes. In this phase, a model was developed using a nonlinear mixed-effects modelling approach, where the site classes were incorporated as a random effect, allowing key model parameters, e.g. asymptote, to vary among site classes. This approach enabled the model to capture systematic differences in H-D relationships associated with site productivity while simultaneously utilising information from the full dataset to improve parameter estimation and overall model goodness-of-fit.

Phase II involved fitting H-D models for each site class. This phase was based on the assumption that the general mixed-effects H-D model, which treated site class as a random effect, might not fully capture site class-specific H-D relationships with sufficient accuracy when predicting H for each class. By stratifying the data by site class, this approach capitalised on the relatively large sample sizes available within each class and enabled the development of site-class-specific H-D models [14].

The H-D relationships were modelled using biologically meaningful equation forms that describe tree H growth as a function of D. Numerous functional forms for H-D relationships have been documented in the literature [19][20]. From these, two well-established nonlinear models (Equations (2) and (3)), that have been applied elsewhere and performed well, were selected for this study [21][22]. These models were chosen for their proven flexibility and their ability to represent the sigmoidal growth pattern characteristic of tree H development, characterised by rapid H increment at smaller diameters followed by a gradual levelling off as stands mature. In these formulations, parameter β 0 determines the asymptotic maximum H approached as diameter increases, while β 1 governs the rate at which H approaches this asymptote, thereby controlling the curvature and overall shape of the H-D relationship [23].

where β ′ s are model parameters to be estimated.

When fitting the generalised H-D models, the random effects were allowed to vary across the parameters β 0 and/or β 1 . The parameters β 0 and β 1 were expressed as β 0 + S₀ and β 1 + S_i, respectively, where S ′ s expresses the difference of parameter β 0 and β 1 of site class i from the mean value obtained from Equations (2) or (3) or the typical site class. All H-D models were fitted using nlme package in R software [24]. The random functions in the nlme package were not enabled when fitting site-class-specific models. In both modelling approaches, a power variance function (VarPower) was applied to account for heteroscedasticity in the residuals [25]. Models’ performance was evaluated using root mean square error (SE), coefficient of determination (R²), mean prediction error (PE%, Equation (4)), and the Akaike Information Criterion (AIC). Although all metrics were considered to assess goodness of fit and predictive accuracy, final model selection was primarily based on the model with the lowest AIC. The significance of PE% was further assessed using a paired t-test, comparing the observed H and the estimated H.

where PE% is the mean prediction error percentage, H i is the observed value of total tree height for observation i, H ^ i is the predicted value of total tree height for observation i.

2.5.2. Validation of the Selected Models

Model validation was conducted using a repeated-random holdout approach. For each iteration, 80% of the observations were randomly assigned to the training dataset for model calibration, while the remaining 20% were reserved for independent validation [26]. This validation procedure was applied consistently to both the generalised H-D models and the site class-specific H-D models. To reduce the influence of random partitioning and ensure stable validation results, the resampling process was repeated 50 times. Performance metrics, including the mean standard error percentage ( SE % = 100 ∗ SE H ¯ ) and the PE%, were averaged across all iterations, and the corresponding confidence intervals were computed to quantify uncertainty and assess model reliability.

The performance of the fitted H-D models is presented in Table 2. For the general H-D equations, the R² ranged from 0.71 to 0.77, while SE varied between 2.83 and 3.16. The mixed-effects general model exhibited a higher R² (0.76 and 0.77) and lower SE (2.83) than the corresponding fixed-effects model (R² = 0.71; SE = 3.15). Incorporating site index as a random effect improved model fit relative to the non-mixed model, as evidenced by a lower AIC value. Among the two model forms evaluated (Equations (2) and (3)), Equation (3) fitted with mixed effects was selected as the best-performing general H-D model. None of the PE% values were significantly different from zero (p > 0.05) in all cases.

For the site-quality-specific models, the R² ranged from 0.70 for site classes I and II to 0.81 for site class III. The SE for site classes I and III were identical across the competing models; however, for site class II, Model 2 exhibited a lower SE. In addition, PE% values were consistently low compared with the general H-D models and were not significantly different from zero (p > 0.05). Consequently, Model 2 was selected for site classes I and II, and Model 3 for site class III. Model expressions of the selected model are presented in Equations (5) to (8).

The patterns of the selected H-D models’ curves, along with the site-class scatter plots, are shown inFigure 3. As expected, the model’s asymptotic values increased with improving site quality. The estimated asymptotic H values were 27.4 m for site class I, 20.4 m for site class II, and 16.9 m for site class III. The selected general H-D model yielded an estimated asymptotic H of 23.4 m, which was well below site classes I and II but higher than site class III. The residual plots of the selected H-D models are shown in Figure 4. The residual plots for all selected models did not indicate any bias.

Table 2.Performance of the fitted H-D models.

Standard error of parameters in their order: 0.4803; 0.0017; and 0.0221

Standard error of parameters in their order: 0.2493; 0.0031; and 0.0474

Standard error of parameters in their order: 0.2493; 0.0031; and 0.0474

Standard error of parameters in their order: 0.2280; 0.0025; and 0.0388.

Variance component of the random effect: variance of the asymptote parameter 3.209973*10⁻⁸; STD = 0.00018; variance of Residual = 0.083.

The validation results for the selected H-D models are presented in Table 3. For the general H-D models, both the SE% and the PE% exhibited a clear dependence on site quality. Model performance was strongest in site class II, while reduced accuracy was observed in both site class I and site class III. Across the site classes, SE% values ranged from 15.89% to 25.49%, whereas PE% varied from −3.01% to −25.47%. These results indicate that the general H-D model provides relatively accurate height predictions under moderate site conditions but performs poorly at the extremes of site quality, where prediction uncertainty and bias increase substantially.

Figure 3.Site classes’ scatter plots and the curves of the selected site classes and general H-D models.

Figure 4.Residual plots of the selected H-D models.

This pattern is further illustrated in Figure 5, which shows the distribution of PE% across validation iterations for each site class when applying the site quality-specific models. The PE% remains relatively stable and near zero, indicating consistent and unbiased model performance when applying the site class specific models. In contrast, PE% increases progressively as site quality deviates from site class II, either toward higher or lower site classes, with the largest deviations observed in site class III when applying the general H-D models. This trend highlights the reduced reliability of the general H-D model under site conditions that depart from moderate site quality. When assessing the model on the whole data set, the SE% and PE% were found to be lower.

The site-specific H-D models consistently exhibited stable, superior performance across all site classes. The SE% values remained within a narrow range of 15.62% to 17.32%, while the PE% ranged from −3.39% to −3.83%, indicating minimal bias when the site quality-specific model is applied to the corresponding site quality. This stability is also evident in Figure 5, where the magnitude and spread of PE% across validation iterations are comparatively small and uniform for the site-specific models. Overall, these findings highlight the advantage of using site-specific H-D models, as they substantially reduce prediction error and bias and provide more reliable height estimates across varying site conditions.

Table 3.Validation results showing the standard errors and mean prediction errors of selected models, tested on independent data.

*CI: Confidence Interval of the mean value.

Figure 5. The pattern of the prediction error percentage across the validation iterations.