Allometric Models to Estimate the Merchantable Wood Volume and Biomass of the Most Abundant Miombo Species in the Miombo Woodlands in Mozambique

Abstract

The Miombo woodlands are declining in both area and value, primarily due to over-harvesting of commonly preferred species. These forests, however, still contain several other species that are potentially of commercial importance. This study aimed to address the need for improved volume and biomass estimates for the sustainable management and utilization of two of the most abundant timber species in Mozambique’s Miombo woodlands: Brachystegia spiciformis (common name: Messassa) and Julbernardia globiflora (common name: red Messassa). Non-linear models were developed to estimate the merchantable wood volume under bark, heartwood volume, and biomass. The volume and biomass models for wood and heartwood volume, which included both diameter at breast height (DBH) and tree height as predictor variables, outperformed single-predictor models. However, the performance of some ratio models using DBH as the only predictor variable surpassed that of models using two predictor variables. The developed models are recommended for adoption by forest companies to increase economic and environmental benefits as they can refine harvest planning by improving the selection of trees for harvesting. Proper tree selection enhances the rate of recovery of high-quality timber from heartwood while observing sustainable forest management practices in Miombo and increasing the proportion of carbon removed from forests, which is subsequently stored in wood products outside the forest.

1. Introduction

The Miombo woodlands are the largest tropical seasonal woodland and dry forest found in Africa [1], covering an area of 1.9 million km² [2]. The Miombo woodlands play a crucial role in climate regulation and the socio-economic development of many sub-Saharan countries by sequestering carbon and helping households absorb, withstand, and recover from climatic or economic shocks [3,4].

In Mozambique, Miombo woodland is the most important forest type, since it covers three-thirds of the total forest area of the country, which is estimated to be 34 million hectares [5,6]. These forests support the livelihood of over 70% of Mozambique’s population, contributing approximately 20% to household cash income, 40% to household subsistence (non-monetary), and 80% to the country’s total energy consumption [6,7]. Additionally, the Miombo woodlands contribute significantly to the country’s economy, accounting for about 13.7% of the Gross Domestic Product (GDP) in 2016, generating up to 214,000 jobs, and storing approximately 5.2 trillion tCO₂eq [6,7]. Therefore, it is necessary to responsibly manage this forest formation.

In Mozambique, Brachystegia spiciformis (common name: Messassa) and Julbernardia globiflora (common name: Red Messassa) are among the most abundant species, yet they are less known [5,8,9]. Notably, the national annual allowable cut (AAC) for these two species (599,296 m³/year) accounts for 31% of the country’s total AAC, whereas the AAC for the three most harvested species—Pterocarpus angolensis, Combretum imberbe, and Afzelia quanzensis—is only 96,713 m³/year, representing just 5% of the total AAC [10]. Additionally, Brachystegia spiciformis and Julbernardia globiflora are among the few timber species with commercially viable populations [8]. Furthermore, some researchers argue that the commercial value of these two species should be increased, as their physical and chemical properties, along with their workability, are comparable to those of high-value species [8,9,11]. Therefore, promoting their use could play a crucial role in enhancing the sustainable management of Miombo forests by alleviating pressure on the most valuable and consequently most exploited timber species [8,9]. However, to achieve this, in addition to conducting studies aimed at understanding their properties and identifying suitable uses, it is essential to develop mechanisms that enable the accurate, rapid, and cost-effective estimation of different volume categories, their ratios, and biomass at the individual tree level for these species at a local scale.

Among the measurable categories of volume and biomass, the merchantable volumes of the stem, branches, and heartwood are particularly significant [12,13,14]. These metrics provide essential insights into the actual wood volume available for industrial applications, enhance qualitative assessments of harvested timber, inform the selection process for trees designated for harvesting, and facilitate accurate evaluations of logging impacts on carbon emissions, as well as the potential for carbon sequestration in wood products with varying lifespans [12,13,14]. Their measurement at different time points also offers valuable data for growth and yield studies related to these various merchantable components [15]. Additionally, predicting the ratios of component volumes to stem or total merchantable volume can improve the tree selection process for felling, increase the recovery rate and the proportion of harvested wood that will be incorporated into wood products, reduce operational costs, and improve the quality of the harvested timber, factors that optimize the profitability of logging activities and reduce the impact of logging on carbon emissions [16,17].

To assess the impact of logging on forest carbon stocks, it is essential to obtain accurate estimates of the biomass removed from forests and incorporated into wood products [14,18,19]. Utilizing allometric equations to estimate the merchantable volume of harvested trees enhances the precision of these estimates, enabling a more accurate evaluation of how much of the carbon previously stored in the forest will potentially be converted into carbon stocks in wood products outside the forest [14,20,21]. This enhances the identification of viable business models that encompass timber products, co-products, and carbon credits [18,22]. This step is essential for optimizing wood utilization by considering conversion rates and increasing wood valuation, as well as for evaluating the climatic impact of using wood relative to other materials in goods production and infrastructure development [23,24].

Allometric equations have been developed for estimating the total tree wood volume and biomass for mixed Miombo species by [25,26] and for the most valuable Miombo species by [27], as well as for the determination of tree components’ wood volume, such as merchantable branches and stems, by [13,28]. However, species-specific equations for lesser-known species are scarce. The main objective of this study is to develop species-specific allometric equations for estimating the total merchantable wood volume, the volumes of tree components and their ratios, and biomass for B. spiciformis and J. globiflora from the Miombo woodlands in Mozambique.

2. Materials and Methods

2.1. Study Area

Data for this study were collected from Levasflor Company forests located in Cheringoma district, Sofala province, central Mozambique (Figure 1). The area is characterized as a rainy tropical savanna on a flat slope with a low altitude (30–200 m.a.s.l.) [29]. The average annual rainfall exceeds 1000–1200 mm, primarily during the rainy season (November–April), and the average annual temperature is 23 °C [29]. The soil is sandy, extremely well drained, and has a low nutrient and water retention capacity [30].

2.2. Data Collection

A total of 37 B. spiciformis trees (DBH ranging from 17 to 81cm and height ranging from 12 to 26 m) and 23 J. globiflora trees (DBH ranging from 14 to 62 cm and height ranging from 12 to 23 m) were sampled. The selected trees were evenly distributed across the available DBH classes. The trees were chosen based on the absence of visible deformities, abnormalities, or diseases that might interfere with their industrial processing, following standard practices employed by forest companies in Mozambique. This approach is commonly applied in selecting sample trees for volume and biomass studies as it helps minimize errors, particularly when the measurements target specific volume/biomass categories rather than the total stand volume/biomass, as demonstrated by [31,32,33].

Once selected, the tree’s DBH, total height, and merchantable height were measured prior to felling. A statistical summary of these variables is provided in

Table 1. Summary and descriptive statistics of field-measured data.

	Brachystegia spiciformis			Julbernardia globiflora
	DBH (cm)	Hc (m)	Ht (m)	DBH (cm)	Hc (m)	Ht (m)
Min	17.1	4.46	12.5	14.5	4.46	12.03
Max	80.7	18.7	25.4	61.2	13.9	22.17
Mean	47.04	10.26	19.92	38.93	9.17	18.28
SD	20.62	3.92	3.42	15.71	3.17	2.77
CV	43.84	38.22	17.18	40.34	34.55	15.15
SE	3.39	0.65	0.56	3.28	0.66	0.58

. In addition, after felling, each tree was cross-cut, and its diameter under and over bark and the diameter of the heartwood stem were measured at lengths of 0.3 m, 0.8 m, 1.5 m, and 2.3 m and then in sections every 3 m up to a top diameter over bark of 10 cm. The branches were trimmed and cross-cut into 1 m sections from the bottom up to a top diameter over bark of 10 cm, and the diameter over bark and under bark and the diameter of heartwood were measured at the ends of each section. Only large branches (top diameter ≥ 10 cm) were measured, and, to comply with the typical practices of forest companies in Mozambique, only branches with at least a length of 1.5 m were selected for developing the models. The flowchart in Figure 2 shows the data collection, processing, and analysis.

2.3. Determination of Volumes, Ratios, and Biomass

The volume of each section of the felled trees was computed using the Smalian formula (Equation (1)).

$$V_m=L\times {\displaystyle \frac{(A_b+A_u)}{2}}$$

(1)

where

• V_m = total volume of the section (under bark and heartwood) (m³);
• A_b = cross-sectional area at the bottom of the section (m²);
• A_u = cross-sectional area at the top of the section (m²);
• L = length of the section (m).

The total merchantable volume and the volume of the components were computed by summing the volumes of the corresponding sections.

Table 2. Categories of volumes and biomass computed in this study.

Category	Wood Component	Tree Part	Analyzed Volume and Biomass Categories
Total wood	Heartwood plus sapwood	Stem plus branches	Total tree merchantable wood volume under bark (stem merchantable wood volume under bark plus branch merchantable wood volume under bark) (stem or branches from 15cm diameter over bark)
			Total tree merchantable biomass under bark (biomass of stem merchantable wood volume under bark plus biomass of branches merchantable wood volume under bark) (stem or branches from 15 cm diameter over bark)
Components	Heartwood and sapwood	Merchantable stem	Stem merchantable wood volume under bark (stem from 15 cm diameter over bark)
		Merchantable branches	Branches merchantable wood volume under bark (branches from 15 cm top diameter over bark)
	Heartwood	Stem plus branches	Total merchantable heartwood volume (stem merchantable heartwood plus branch merchantable heartwood)
		Merchantable stem	Stem merchantable heartwood (stems from 15 cm diameter over bark)
		Merchantable branches	Branches merchantable heartwood (branches from 15 cm diameter over bark)

presents the categories of merchantable volumes and biomass computed in this study.

Two types of ratios were determined: the ratio of the merchantable volume of the stem in relation to the total merchantable volume of the tree under bark (branches from 15 cm diameter over bark), and the ratio of the volume of the heartwood of each component in relation to the volume of the wood (heartwood and sapwood) of the component to which the heartwood belongs.

In the model fitting process for the total and component volumes, identical models were applied to ensure the additivity of the component volumes, as proposed by [34]. This approach ensures that the sum of the component estimates equals the total volume obtained from the total equation [34,35].

The basic wood density of each component (the stem, primary branches, secondary branches, and tertiary branches) was used to compute the merchantable biomass, as in [36]. To determine the basic wood density, seven B. spiciformis trees and six J. globiflora trees, one for each diametric class, were selected. Four 7 cm thick discs were obtained from each tree: one at the DBH position and the others from the first, second, and third branches. Each disc was converted into four wedges; two opposite wedges were selected, and the basic wood density was determined following the [37]. The fresh volume of the wedges was determined via hydrostatic weighing, and the dry weight was recorded after the wedges were dried to a constant mass at 103 ± 2 °C. The basic wood density of each disc was calculated as dry mass/fresh volume.

2.4. Selection of Candidate Models

An exploratory data analysis was carried out through scatter plots to observe patterns in the relationships between DBH and selected dependent variables (the total merchantable volume, total merchantable biomass, and the volume of merchantable components, namely, the branches, stem, and heartwood). Based on the scatter plot patterns, the candidate models shown in

Table 3. Candidate allometric models for merchantable volume and biomass.

Model	Power Models	Model	Ratio Models
1	$Y=b_0\times {DBH}^{b_1}\times$ ε	4	$g\left(\mathsf{\mu }\right)=b_0+b_1\times DBH+$ ε
2	$Y=b_0\times {\left({DBH}^2\times Ht\right)}^{b_1}$ × ε	5	$g\left(\mathsf{\mu }\right)=b_0+b_1\times {DBH}^{2}Ht+$ ε
3	$Y=b_0\times {DBH}^{b_1}\times {Ht}^{b_2}\times$ ε	6	$g\left(\mathsf{\mu }\right)=b_0+b_1\times DBH+b_2\times Ht+$ ε

where DBH: diameter at breast height (cm); Ht: total height (m); b₀, b₁, and b₂: regression coefficients; µ: the mean of link function $g\left(\mu \right)=log\left({\displaystyle \frac{\mu }{1-\mu }}\right)$; ε: random error.

were selected.

As the homogeneity of variance of residuals is an assumption often violated in biological studies that model size [26,38], several procedures have been developed to overcome heterogeneity [35,39]. In the current study, to compensate for the unequal variance of the residual error when fitting non-linear models relating to volume and biomass, the variance was weighted as in [26]. This procedure was used due to its flexibility and its recognized robustness over non-linear models with additive error and log-transformed models [35,40].

For each model (power and ratio), residuals were analyzed using distribution patterns [41] and the Breusch–Pagan test [42,43]. This combination evaluates homoscedasticity by visually identifying trends in residuals and statistically confirming variance consistency, ensuring robust model performance assessment [41,42].

2.5. Model Evaluation and Validation

The most suitable models were selected using the following criteria, listed in order of relative importance: Akaike’s Information Criterion (AIC), the Bayesian Information Criterion (BIC), the Mean Absolute Error (MAE), the Mean Absolute Percentage Error (MAPE), and the Root Mean Squared Error (RMSE) [44]. AIC and the BIC were used to assess model fit while penalizing complexity, with lower values indicating better model performance [45,46,47]. AIC prioritizes goodness of fit with a moderate penalty for additional parameters, whereas the BIC applies a stricter penalty, favoring simpler models. The combined use of AIC and the BIC aimed to balance overfitting (favored by AIC) and under fitting (favored by the BIC), ensuring a more robust model selection while maintaining the principle of selecting models that are as simple as possible and as complex as required [45,46]. If both criteria select the same model, it provides strong evidence of its adequacy.

The MAE measures the average absolute difference between observed and predicted values, offering an intuitive measure of prediction accuracy [48,49,50]. The MAPE expresses these errors as a percentage, enabling comparison across different datasets [49]. The RMSE, which gives higher weight to larger errors, captures overall model accuracy while being more sensitive to extreme deviations [48,50]. The simultaneous use of the MAE and MAPE allowed for a comprehensive assessment of both absolute (MAE) and relative (MAPE) error magnitudes, contributing to a more robust decision-making process. The inclusion of the RMSE helped identify large errors that could compromise prediction quality [48,49,50]. Overall, the combination of these criteria enhanced the reliability of model selection by ensuring a balance between goodness of fit and predictive accuracy. R² was not used because it is inappropriate for non-linear models [51,52]. The AIC, BIC, MAE, MAPE, and RMSE were selected due to their widespread application in studies of this nature conducted in Miombo by the authors of [26,53] and in other forest formations [54,55].

Validation was performed for the selected models for volume, ratio, and biomass using repeated K-fold cross-validation with 10 folds and three repetitions, as suggested in [56,57]. In this analysis, we computed the MAPE and assessed its variability and consistency in relation to the same parameter determined for the models, which was obtained randomly in different sub-samples. Through these evaluation and validation techniques, the accuracy, reliability, and predictive capabilities of the selected model were assessed in other samples [56,57].

2.6. Comparison with Existing Allometric Equations

In this study, the performance of the best-fitting model was compared with that of existing models described in the literature, prioritizing comparisons with models developed for species and climatic conditions similar to those considered in the present study. This procedure is common in modeling and aims to validate and justify the adoption of new models compared to pre-existing ones, as in [25,28]. The comparison assesses the relevance of adopting existing general volume and biomass models with reasonable performance rather than developing new local-species-specific models. In this study, this comparison was restricted to volume and biomass models.

In this context, in terms of the total tree merchantable wood volume under bark, the models developed for the two species in the current study were compared with each other, with the model developed for Holoptelea grandis (Hutch.) Mildbr and Cynometra megalophylla Harms by the authors of [58], and with models developed for mixed species in the Miombo ecoregion stands by the authors of [26,59]. For the stem merchantable wood volume under bark, the models for the species considered in the current study were compared with each other, with the models developed by the authors of [60] for Afzelia quanzensis and Millettia stuhlmannii, and with models for the stem volume of mixed species in the Miombo stand developed by the authors of [13,26]. For the branch merchantable wood volume under bark, the models the species in the current study were compared with each other and with the models developed for the branch volume of mixed species in the Miombo stand by the authors of [10,26,28].

For the total merchantable heartwood volume, the models for the species in this study were compared with each other and with the models developed by the authors of [61,62] for the total heartwood volume of Tectona grandis. The models for the total tree biomass for the species in the current study were compared with each other and with species-specific models developed by the authors of [53,60] for the biomass of Afzelia quanzensis, Pterocarpus angolensis, and Brachystegia spiciformis, as well as with biomass models for mixed species in the Miombo ecoregion stands [25]. No models were found that were considered viable for comparison at the branch (heartwood) and ratio levels.

3. Results

3.1. Allometric Equations for Volume Estimation

The residual scatterplots combined with the p-value of the Breusch–Pagan test for the variables related to the merchantable wood volume under bark and heartwood for both B. spiciformis and J. globiflora indicate homoscedasticity in all cases (Figure 3 and Figure 4). Therefore, the selection of the most suitable merchantable volume models was based exclusively on the AIC, BIC, MAE, MAPE, and RMSE criteria.

Among the three evaluated models for all dependent variables (total tree merchantable wood volume under bark, stem merchantable wood volume under bark, branch merchantable wood volume under bark, total merchantable heartwood volume, stem merchantable heartwood volume, and branch merchantable heartwood volume), Model 2, which incorporates height as a variable (DBH²Ht), demonstrated higher performance for both B. spiciformis and J. globiflora, as per their AIC and BIC values, which were the lowest. Model 1, which relied solely on DBH as the predictor variable, exhibited the poorest performance, with the highest AIC and BIC values. Model 3, which includes both DBH and height (with height as a second variable), performed better than Model 1 but did not outperform Model 2 in overall accuracy (

Table 4. Statistical performance of merchantable volume models for B. spiciformis.

ID	Models	AIC	BIC	RMSE	MAE	MAPE (%)
	Total tree merchantable wood volume under bark
1	Y = 1.67 × 10−4 × DBH2.319	−33.017	−21.741	0.200	0.128	10.631
2	Y = 2.59 × 10−5 × (DBH2 × Ht)1.008	−57.394	−46.117	0.192	0.115	7.411
3	Y = 2.42 × 10−5 × DBHb2.002 × Ht1.047	−49.439	−31.719	0.194	0.116	7.365
	Stem merchantable wood volume under bark
1	Y = 2.03 × 10−4 × DBH2.181	−51.701	−40.425	0.155	0.109	11.680
2	Y = 3.32 × 10−5 × (DBH2 × Ht)0.953	−72.116	−60.840	0.142	0.083	8.111
3	Y = 2.75 × 10−5 × DBH1.873 × Ht1.058	−64.373	−46.653	0.143	0.084	8.077
	Branch merchantable wood volume under bark
1	Y = 1.17 × 10−5 × DBH2.682	−123.212	−111.935	0.078	0.051	12.187
2	Y = 1.58 × 10−6 × (DBH2 × Ht)1.150	−145.192	−133.915	0.091	0.048	8.832
3	Y = 1.78 × 10−6 × DBH2.327 × Ht5.130	−137.316	−119.596	0.089	0.047	8.895
	Total merchantable heartwood volume
1	Y = 7.99 × 10−7 × DBH3.500	−98.802	−87.525	0.135	0.075	11.649
2	Y = 4.36 × 10−8 × (DBH2 × Ht)1.528	−117.166	−105.890	0.118	0.070	9.408
3	Y = 1.13 × 10−7 × DBH3.198 × Ht1.035	−115.792	−98.071	0.101	0.061	9.080
	Stem merchantable heartwood
1	Y = 9.39 × 10−7 × DBH3.391	−112.714	−101.438	0.113	0.068	12.657
2	Y = 5.78 × 10−8 × (DBH2 × Ht)1.478	−130.199	−118.923	0.114	0.063	9.380
3	Y = 1.29 × 10−7 × DBH3.081 × Ht1.053	−126.200	−108.480	0.103	0.057	9.529
	Branch merchantable heartwood
1	Y = 2.83 × 10−8 × DBH43.977	−184.658	−173.382	0.082	0.045	18.862
2	Y = 1.12 × 10−9 × (DBH2 × Ht)1.729	−191.102	−179.826	0.067	0.038	16.989
3	Y = 3.65 × 10−9 × DBH3.649 × Ht1.099	−185.938	−168.218	0.069	0.040	16.820

Akaike’s Information Criterion (AIC), Bayesian Information Criterion (BIC), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), Root Mean Squared Error (RMSE), diameter at breast height (DBH), and total height (Ht).

and

Table 5. Statistical performance of merchantable volume models for J. globiflora.

ID	Models	AIC	BIC	RMSE	MAE	MAPE (%)
	Total tree merchantable wood volume under bark
1	Y = 6.53 × 10−5 × DBH2.737	−2.588	5.360	0.342	0.204	13.502
2	Y = 6.98 × 10−6 × (DBH2 × Ht)1.196	−16.500	−8.552	0.230	0.160	10.474
3	Y = 3.55 × 10−6 × DBH2.286 × Ht1.559	−9.560	2.931	0.219	0.156	10.135
	Stem merchantable wood volume under bark
1	Y = 7.05 × 10−5 × DBH2.627	−10.365	−2.417	0.277	0.169	15.070
2	Y = 8.34 × 10−6 × (DBH2 × Ht)1.1470	−20.879	−12.931	0.204	0.153	12.802
3	Y = 3.02 × 10−6 × DBH2.134 × Ht1.694	−14.382	−1.892	0.191	0.145	12.193
	Branch merchantable wood volume under bark
1	Y = 5.41 × 10−6 × DBH3.064	−50.336	−42.387	0.129	0.089	18.202
2	Y = 4.72 × 10−7 × (DBH2 × Ht)1.333	−56.293	−48.344	0.112	0.075	15.003
3	Y = 3.73 × 10−7 × DBH2.624 × Ht1.464	−48.346	−35.855	0.112	0.075	14.908
	Total merchantable heartwood volume
1	Y = 1.86 × 10−11 × DBH6.337	−89.882	−81.934	0.391	0.198	19.621
2	Y = 1.28 × 10−13 × (DBH2 × Ht)2.748	−120.782	−112.834	0.350	0.173	11.342
3	Y = 2.67 × 10−13 × DBH5.639 × Ht2.325	−116.228	−103.737	0.343	0.176	11.359
	Stem merchantable heartwood
1	Y = 1.88 × 10−11 × DBH6.261	−100.158	−92.210	0.284	0.149	21.231
2	Y = 1.29 × 10−13 × (DBH2 × Ht)2.722	−126.579	−118.630	0.263	0.131	12.303
3	Y = 1.38 × 10−13 × DBH5.457 × Ht2.682	−118.595	−106.105	0.263	0.131	12.354
	Branch merchantable heartwood
1	Y = 1.88 × 10−12 × DBH6.568	−147.223	−139.275	0.116	0.051	20.823
2	Y = 1.09 × 10−14 × (DBH2 × Ht)2.847	−153.325	−145.376	0.099	0.047	17.715
3	Y = 5.17 × 10−14 × DBH5.950 × Ht1.999	−146.343	−133.853	0.098	0.048	17.835

Akaike’s Information Criterion (AIC), Bayesian Information Criterion (BIC), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), Root Mean Squared Error (RMSE), diameter at breast height (DBH), and total height (Ht).

).

3.2. Ratio Models

The residual scatterplots, combined with the p-value of the Breusch–Pagan test for the ratios, indicate heteroscedasticity only in Model 2 for the ratio of stem merchantable wood volume under bark to total tree merchantable wood volume under bark for B. spiciformis (Figure 5). However, for J. globiflora, heteroscedasticity was observed in two models: Model 2 for the ratio of total merchantable heartwood volume under bark to total tree merchantable wood volume under bark and in Model 2 for the ratio of stem merchantable heartwood volume under bark to total tree merchantable wood volume under bark (Figure 6). The models that showed heteroscedasticity are not candidates for the best model. The selection of the best model considered the AIC, BIC, MAE, MAPE, and RMSE criteria for those models remaining contenders for best model.

Model 4, with one predictor variable (DBH), was the best model for estimating the ratio of stem merchantable wood volume under bark to total tree merchantable wood volume under bark for both species, as well as the ratio of total merchantable heartwood volume to total tree merchantable wood volume under bark for B. spiciformis (

Table 6. Statistical performance of the selected models for ratios of B. spiciformis.

ID	Model	AIC	BIC	RMSE	MAE	MAPE (%)
	Ratio of stem merchantable wood volume under bark vs. total tree merchantable wood volume under bark
4	Y = 1.533 − 0.012 × DBH	−187.216	−182.383	0.018	0.015	2.128
5	Y = 1.261 − 4.9 × 10−6 × (DBH2 × Ht)	−183.582	−178.749	0.019	0.015	2.129
6	Y = 1.656 − 0.011 × DBH − 0.009 × Ht	−184.099	−174.433	0.018	0.014	2.029
	Ratio of total merchantable heartwood volume vs. total tree merchantable wood volume under bark
4	Y = −2.452 + 0.037 × DBH	−120.798	−115.965	0.045	0.036	13.392
5	Y = −1.543 + 1.52 × 10−5 (DBH2Ht)	−99.164	−94.331	0.058	0.048	19.468
6	Y = −2.834 + 0.034 × DBH + 0.027 × Ht	−117.079	−107.413	0.043	0.034	12.733
	Ratio of stem merchantable heartwood vs. stem merchantable wood volume under bark
4	Y = −2.893 + 0.063 × DBH	−84.558	−79.726	0.073	0.059	14.469
5	Y = −1.498 + 2.82 × 10−5 × (DBH2Ht)	−93.142	−88.309	0.066	0.053	15.298
6	Y = −2.679 + 0.058 × DBH − 0.003 × Ht	−88.089	−78.424	0.072	0.060	14.290

Akaike’s Information Criterion (AIC), Bayesian Information Criterion (BIC), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), Root Mean Squared Error (RMSE), diameter at breast height (DBH), and total height (Ht).

and

Table 7. Statistical performance of the selected models for ratios of J. globiflora.

ID		AIC	BIC	RMSE	MAE	MAPE (%)
	Ratio of stem merchantable wood volume under bark to total tree merchantable wood volume under bark
4	Y = 1.497 − 0.013 × DBH	−74.966	−71.560	0.042	0.034	4.708
5	Y = 1.250 − 7.14 × 10−6 × (DBH2 × Ht)	−72.716	−69.310	0.044	0.035	4.857
6	Y = 1.146 − 0.016 × DBH + 0.005 × Ht	−74.667	−67.854	0.043	0.034	4.825
	Ratio of total merchantable heartwood volume to total tree merchantable wood volume under bark
4	Y = −5.892 + 0.098 × DBH	−97.364	−93.957	0.048	0.030	43.407
5	Y = −3.728 + 5.18 × 10−5 (DBH2Ht)	−82.850	−79.444	0.058	0.040	104.198
6	Y = −8.505 + 0.094 × DBH + 0.145 × Ht	−106.714	−99.901	0.056	0.034	21.837
	Ratio of stem merchantable heartwood to stem merchantable wood volume under bark
4	Y = −6.061 + 0.166 × DBH	−71.651	−68.244	0.071	0.046	39.758
5	Y = −3.564 + 6.40 × 10−5 × (DBH2Ht)	−69.126	−65.720	0.073	0.054	92.633
6	Y = −8.566 + 0.110 × DBH − 0.140 × Ht	−93.239	−86.426	0.076	0.049	20.017

Akaike’s Information Criterion (AIC), Bayesian Information Criterion (BIC), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), Root Mean Squared Error (RMSE), diameter at breast height (DBH), and total height (Ht).

), since it observed the lowest AIC and BIC values. Model 5 demonstrated higher performance, for B. spiciformis, in computing the ratio of stem merchantable heartwood to stem merchantable wood volume under bark, as it has the lowest AIC and BIC values, the smallest RMSE and MAE. Model 6 exhibited the best performance for the ratio of total merchantable heartwood volume to total tree merchantable wood volume under bark and for the ratio of stem merchantable heartwood to stem merchantable wood volume under bark for J. globiflora, showing the lowest AIC and BIC values and the smallest MAPE.

3.3. Biomass Equations

Basic wood density was used to estimate biomass. The mean density of wood (g/cm³) and related standard deviation values at the DBH position, first branch, secondary branch, and tertiary branch were 0.575 ± 0.012, 0.563 ± 0.022, 0.562 ± 0.011, and 0.551 ± 0.009 for B. spiciformis, respectively, and 0.699 ± 0.012, 0.687 ± 0.014, 0.684 ± 0.008, and 0.679 ± 0.007 for J. globiflora, respectively. There was no evidence of a statistically significant difference in density among the established positions for B. spiciformis (p = 0.076). However, a significant difference was observed for J. globiflora (p = 0.036), particularly between the density of tertiary branches and the density at the DBH position. A significant difference was also observed between the two species (p = 2.53 × 10⁻³⁰).

The residual scatterplots combined with the p-value of the Breusch–Pagan test for the models for both species of the total tree merchantable biomass and the total merchantable wood under bark indicate homoscedasticity (Figure 7 and Figure 8). Therefore, the selection of the best model was based on the AIC, BIC, MAE, MAPE, and RMSE criteria.

Table 8. Statistical performance of biomass models for B. spiciformis.

ID	Models	AIC	BIC	RMSE	MAE	MAPE (%)
	Total tree merchantable biomass of the total merchantable wood under bark
1	Y = 8.83 × 10−5 × DBH2.343	−74.835	−63.558	0.111	0.071	10.657
2	Y = 1.35 × 10−5 × (DBH2 × Ht)1.017	−98.324	−87.048	0.112	0.069	7.682
3	Y = 1.33 × 10−5 × DBH2.030 × Ht1.028	−90.328	−72.608	0.112	0.069	7.670

Akaike’s Information Criterion (AIC), Bayesian Information Criterion (BIC), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), Root Mean Squared Error (RMSE), diameter at breast height (DBH), and total height (Ht).

and

Table 9. Statistical performance of biomass models for J. globiflora.

ID	Models	AIC	BIC	RMSE	MAE	MAPE (%)
	Total tree merchantable biomass of the total merchantable wood under bark
1	Y = 4.10 × 10−5 × DBH2.763	−20.095	−12.147	0.237	0.138	13.250
2	Y = 4.29 × 10−6 × (DBH2 × Ht)1.207	−34.244	−26.295	0.237	0.110	10.298
3	Y = 2.26 × 10−6 × DBH2.314 × Ht1.551	−27.216	−14.726	0.151	0.108	9.995

Akaike’s Information Criterion (AIC), Bayesian Information Criterion (BIC), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), Root Mean Squared Error (RMSE), diameter at breast height (DBH), and total height (Ht).

show that, for both species, Model 2, which uses height as an additional predictor variable (D²H), demonstrated the best performance in terms of AIC and the BIC. Conversely, Model 1 exhibited the poorest performance, with the highest AIC, BIC, MAE, and MAPE values. Although Model 3 outperformed Model 1, it did not surpass Model 2 in overall accuracy.

3.4. Cross-Validation of Volume and Ratio Biomass Equations

For the volume equations for B. spiciformis, the mean MAPE values ranged from 18.03 to 7.80, with higher values observed for the branches and heartwood, while, for J. globiflora, the values of the same criteria ranged from 19.84 to 11.66, with higher values observed for the same components (

Table 10. Mean MAPE values from cross-validation for volume parameters ¹.

Species	Parameter	TWuB	SWuB	BWuB	THwV	SHwV	BHwV
B. spiciformis	MAPE (%)	7.80	8.55	9.31	9.88	9.96	18.03
J. globiflora	MAPE (%)	11.66	13.96	16.33	13.09	14.15	19.84

¹ Total merchantable wood volume under bark (TWuB), stem merchantable wood volume under bark (SWuB), branch merchantable wood volume under bark (BWuB), total merchantable heartwood volume (THwV), stem merchantable heartwood (SHwV), branch merchantable heartwood (BHwV), and Mean Absolute Percentage Error (MAPE).

). The average MAPE for the cross-validation of biomass in the equations was 11.04% and 8.03% for B. spiciformis and J. globiflora, respectively.

For the ratio equations of B. spiciformis, the mean MAPE values ranged from 16.05 to 2.23, with higher values observed for ratios involving heartwood, while, for J. globiflora, the values of the same criteria ranged from 23.01 to 4.93, with higher values observed for the same components (

Table 11. Mean MAPE values from cross-validation for ratio parameters ¹.

Species	Parameter	Ratio SWUB vs. TWUB	Ratio THWV vs. TWUB	Ratio SMHW vs. SWUB
B. spiciformis	MAPE (%)	2.23	14.12	16.05
J. globiflora	MAPE (%)	4.93	25.84	23.01

¹ Ratio of stem merchantable wood volume under bark vs. total tree merchantable wood volume under bark (ratio SWuB vs. TWuB), ratio of total merchantable heartwood volume vs. total tree merchantable wood volume under bark (ratio THwV vs. TWuB), ratio of stem merchantable heartwood vs. stem merchantable wood volume under bark (ratio SHwV vs. SWuB), Mean Absolute Percentage Error (MAPE).

). For the biomass model of B. spiciformis, the mean MAPE value was 8.03, while, for J. globiflora, it was 11.66% (

Table 12. Mean MAPE values from cross-validation for biomass.

Species	Parameter	Biomass
B. spiciformis	MAPE (%)	8.03
J. globiflora	MAPE (%)	11.66

Mean Absolute Percentage Error (MAPE).

).

3.5. Evaluation of the Predictive Performance of Developed and Existing Allometric Equations

All the merchantable volume models developed in this present study, whose dataset served as a basis for evaluating other models, demonstrated superior performance in terms of the MAPE across all variables analyzed for the dataset used (

Table 13. MAPE of the models developed in the present study and in previous studies using volume data from the current study.

References	Models	MAPE (%)
		TWuB	SWuB	BWuB	THwV
Present study	B. spiciformis (data set used)	7	8	9	9
Present study	J. globiflora	81	155	108	159
[58]	Holoptelea grandis	46	-	-	-
[58]	Cynometra megalophylla	29	-	-	-
[59]	Mixed species	86	-	-	-
[26]	Mixed species	82	53	205	-
[60]	Pterocarpus angolensis	-	217	-	-
[60]	Afzelia quanzensis	-	76	-	-
[13]	Mixed species	-	64	98	-
[28]	Mixed species	-	-	334	-
[61]	Tectona grandis	-	-	-	61
[62]	Tectona grandis	-	-	-	100

Total tree merchantable wood volume under bark (TWuB); stem merchantable wood volume under bark (SWuB), branch merchantable wood volume under bark (BWuB), total tree merchantable heartwood (THwV), and Mean Absolute Percentage Error (MAPE).

).

Similarly, all biomass models developed in this study, whose dataset served as the basis for evaluating all other models, demonstrated higher performance in terms of the MAPE across all variables analyzed for the dataset used (

Table 14. MAPE of models developed in the present study and in previous studies using biomass data from the current study.

References	Models	MAPE (%)
		Biomass
Present study	B. spiciformis (data set used)	8
Present study	J. globiflora	166
[53]	B. spiciformis	25
[27]	Pterocarpus angolensis	29
[27]	Afzelia quanzensis	128
	Mixed species	89

Mean Absolute Percentage Error (MAPE).

).

4. Discussion

• Wood volume equations

The total, stem, and branch merchantable wood (heartwood plus sapwood) volume equations of the studied species exhibited MAPE values that fall within the ranges reported in [26] (0.26–18.9%) and [28] (0.0–16.6%) in Tanzanian Miombo woodlands, while the RMSE values were lower than that reported in [59] (0.315 m³–0.47 m³), [26] (0.36–1.10 m³), and [59] (0.142–0.355 m³), studies conducted in the Tanzanian Miombo woodlands. The models in [26,28] were developed for mixed species, while the authors of [59] developed species-specific equations for Brachystegia spiciformis, Combretum molle, and Dalbergia arbutifolia.

The MAPE values obtained in this study were consistently slightly higher for the equation for stem merchantable wood volume under bark than for that of the total tree merchantable wood volume under bark for J. globiflora. This observation is consistent with the findings of [13,26,28], where total wood volume models performed better than stem wood volume models. That stem models achieve the lowest performance may be explained by the fact that determining the merchantable stem height often involves some degree of subjectivity due to imprecision in defining the upper point for measuring the stem height [26]; according to the same authors, this can be attributed to factors such as the variability in first branching and the appearance of defects in the stem.

In general, models with only one predictor variable (DBH) exhibited lower performance than those with two variables. This observation is in agreement with the findings reported by the authors of [13,26,28], when modeling wood volume in Miombo woodlands. The inclusion of the total height variable improved model performance. However, the improvement was not very significant because, as proposed in [26,63,64], DBH and total height are related. Authors, such as those of [13,26,28], also observed that including height in a given model with an existing DBH variable neither adds significant new information nor improves the model performance in Miombo woodlands. Nevertheless, incorporating height as a combined variable with DBH (DBH²Ht) resulted in better-performing models compared to those in which height was included as a second variable, as was found in [28].

It is generally observed that the equations of branches exhibited the highest MAPE values when compared to those of the total and stem volume. Other studies that modeled the total volume, stem volume, and branch volume also reported low performance for models of branches in Miombo woodlands [13,26,28]. The lower accuracy of allometric equations for branches can be attributed to the greater plasticity of branches compared to stems [65,66]. This difference arises from the more variable anatomical structure of branches relative to trunks [65,66,67]. Branches exhibit adaptive growth patterns in response to environmental conditions, whereas trunks are primarily designed for transport, stability, and support, resulting in a more uniform structure that emphasizes strength over adaptability [66,67]. This is particularly evident in the ability of tree species to develop different crown shapes in open and closed environments in response to variations in environmental conditions [66,67,68].

• Heartwood volume equations

The MAPE values of the heartwood equations for the studied species tended to be slightly higher than those of total wood volume equations (sapwood plus heartwood), and the MAE values were higher than those found by the authors of [62] when modeling the total heartwood volume of Tectona grandis in a plantation in China (0.0019 m³). On the other hand, the authors of [69] reported an RMSE value of 0.125 m³ when modeling total heartwood in Eucalyptus globoidea, which is close to the range of values presented in this study. The differences in the MAE and RMSE values may be attributed, among other factors, to the variables that influence heartwood formation in tropical species [70,71]. These include the sociological position of the crown, trunk size, growth rate, species or genera characteristics, and local agro-ecological conditions [70,71,72]. Silvicultural treatments such as spacing, thinning, pruning, fertilization, and irrigation also play a significant role [70,73]. Furthermore, aspects such as metabolism, hormonal regulation, transcriptional control, and cellular biology also influence heartwood formation [62].

The heartwood merchantable volume models suggest that the total merchantable heartwood volume and stem merchantable heartwood volume of both studied species increase exponentially with the increment of DBH, as was observed in [61,74]. However, the authors of [75] observed in their study of the total heartwood volume in teak trees the increase in the heartwood volume with increasing DBH was not exponential but polynomial.

• Ratio models

The relevance of the ratio lies in its ability to quickly provide insights into the proportion of wood with properties and characteristics suitable for processing, as well as the proportion that could potentially be allocated to less valuable purposes (branch wood) [76,77]. By using Beta regression models to model ratios, it was possible to conclude that a high proportion of merchantable branch volume is expected from trees in larger diameter classes. According to the ratio models developed in this study, the increase in DBH results in a reduction in the proportion of the stem merchantable wood volume to the total tree merchantable wood volume. On the other hand, increasing DBH also produces an increase of the proportion of the stem merchantable heartwood volume to the stem merchantable wood volume; the same patterns were found by the authors of [74,75] when modeling the proportion of total heartwood to the total wood volume of teak trees (Tectona grandis). This pattern means that higher proportions of heartwood are associated with trees of larger diameter, indicating that fewer large trees would be needed to produce a given heartwood stem volume required for high-quality products. Although it is rarely addressed in the literature, it is worth noting that ratios should be modeled at the species and site levels, as different allometric relations that can affect ratio models (heartwood/total wood as a function of DBH) have been identified among models developed in different climatic zones [74].

• Practical use of volume and ratio models

Combining information that can be obtained from the wood, heartwood, and ratio models developed in this study can aid in the adoption of suitable strategies for selecting trees for harvesting according to specific objectives. The wood and heartwood merchantable volume models discussed above show increments of the stem merchantable wood volume and stem heartwood volume as DBH increases. This means that fewer larger trees are needed to produce a given volume of wood and heartwood from the stem. However, considering the behavior of the ratio models discussed above, a high volume of merchantable branches is expected from large-diameter trees. Therefore, the decision to select trees for a specific purpose should be made considering that the expected volume of merchantable branches decreases when using a greater number of small trees, but a high volume is generated when using a smaller number of large trees. Users can choose the second option if there is a clear strategy for using branches; otherwise, the first option is preferred. Although it is challenging to use branches due to their small size and irregular shape, which lower the recovery rate, they can be valorized by manufacturing small goods of high quality from their small heartwood material [58].

When selecting larger trees, the volume of heartwood is maximized by the higher total amount of wood per stem as well as the greater proportion of heartwood per stem. Users should therefore also prioritize larger trees if they want to produce high-quality products. However, because large trees play a critical role in forest regeneration and maintaining ecosystem quality and value [78,79], some large trees should be left dispersed throughout the stand. This selective retention approach supports ecological stability and biodiversity, balancing economic gains with forest sustainability. Therefore, harvesting larger trees offers both economic and climatic advantages, as the wood of larger trees typically holds higher value and higher carbon content [78,80,81], so harvesting and processing them increases profit and significantly contributes to increasing carbon storage outside the forest in wood products, owing to their high recovery rate.

• Biomass models

The biomass models exhibited higher values for some model evaluation criteria compared to values found in other studies, such as the MAPE reported in [25] (0.1% and 1.5%), which involved mixed models for Miombo woodlands; the MAPE reported in [82] for species-specific models for Julbernardia paniculata, Brachystegia spiciformis (2.89%), Marquesia macroura, and Isoberlinia angolensis and overall (1.35–7.15%) for the Miombo woodlands; and the MAE reported in [25] (21.8 kg and 6.9 kg). The values obtained in the present study are similar to the RMSE values reported by the authors of [83] (0.163 to 146 kg), who worked with mixed models in Amazon, as well as those reported by the authors of [84], who worked with species-specific models for Brachystegia spiciformis (127–162 kg), Combretum molle (170–203 kg), and Dalbergia arbutifolia (150–206 kg) in Miombo woodlands.

The biomass models in this study showed lower evaluation criteria values compared to those reported in other studies. For instance, the authors of [77,85] reported MAPE values of 18.65–29.30% for mixed-species models in a dry Afromontane forest; the authors of [83] reported MAE values of 144–121 kg and an RMSE of 530 kg for mixed models in the Amazon for commercial stems; and the authors of [86] reported RMSE values of 353–617 kg for species-specific models of Diospyros abyssinica (Hiern) F. White. Similarly, the authors of [14] reported an MAE value of 304 kg for mixed models in the Amazon. Due to the fact that the values obtained in the present study, when compared to others found in the literature, are higher than some [25,82], similar to others [83,84], and lower than some [77,85], it can be concluded that the evaluation metrics obtained in this study fall within the range reported in the literature.

Incorporating height into biomass models improved model performance, particularly when height was included as a combined variable (DBH²H), which is consistent with the findings of [87]. This effect of adding height in biomass models was also found by [53]; however, in their study, the best performance was found when height was included as a separate variable. It is worth emphasizing that the improvement due to height inclusion was not substantial, as observed in [25,53,87]. Therefore, given the challenges associated with measuring tree height and the accuracy of models that do not require direct height measurements, it is justifiable to use equations based solely on diameter.

• Model Performance

The cross-validation results for both species indicate a slight increase in MAPE values compared to the original models, without exceeding 2% for both volume and biomass. The values of the present study are lower than the MAPE values obtained in [85], which ranged from about 18% to 29% for mixed-species biomass equations developed for Afromontane forests. The values of the present study are comparable to the MAPE values obtained in [82], which ranged from −6.09 to 1.35% with the cross-validation method applied to estimate biomass (carbon) in Miombo. The models developed in this study exhibit strong predictive performance, which is expected to remain reliable even when applied to datasets beyond those used for their development. The observed 2% difference is within acceptable limits, considering that cross-validated score estimates often overestimate the expected loss due to the reduced size of the training set compared to the full dataset [56,57].

The comparison of volume and biomass models in this study indicates that our models perform better. Specifically, our biomass model exhibits a broader range and higher values than those reported in previous studies: ref. [85] reported values ranging from 19% to 71%, ref. [25] reported values from 2% to 49%, and ref. [88] reported values from 25% to 290%. Similarly, our total volume model shows greater amplitude and higher values compared to [13], which reported a range of 48% to 139%, and [28], which reported a range of 1% to 52%. These high differences in our study may be due to the fact that several models used for comparison with the values of the present study did not refer to merchantable values. This result aligns with the premise advanced in [25,28,38], which states that local equations specific to species and components (stem, branches, merchantable volume, or entire tree) are essential for accurate modeling.

5. Conclusions

This study developed allometric equations for estimating the total, stem, and branch merchantable wood volume, heartwood, and total merchantable biomass and equations for three ratios, namely, (i) the stem merchantable wood volume under bark to the total tree merchantable wood volume under bark, (ii) the total merchantable heartwood volume to the total tree merchantable wood volume under bark, and (iii) the stem merchantable heartwood to the stem merchantable wood volume under bark, of B. spiciformis and J. globiflora, the two most abundant but less used timber species in the Miombo woodlands in Mozambique.

These models demonstrated reasonable performance, as evidenced by the acceptable MAPE, MAE, and RMSE values, and their use can enhance planning by forest companies by enabling the prior selection of trees suitable for harvesting based on easily measurable variables. This approach provides economic benefits to companies while adhering to good forest management practices and increasing the volumes of biomass and carbon stored in end-user products outside forest ecosystems. The development of local models for these species is recommended, as the performance of existing models for this specific site and species is not sufficient.