Introduction

Chest-height trunk diameter (CHD) and height (Ht) are the two main variables measured in forestry inventories, both being required measurements in decision-making in forest plantation management as well as for research on growth modeling, characterization of stand structure, estimation of timber yield and to determine the predominant height with the purpose of evaluating site quality (^{Costas & Rodríguez, 2003}; ^{Huang, Titus, & Wiens, 1992}; ^{Lappi, 1997}).

While height is difficult to measure and prone to measurement errors, diameter is easily, accurately and economically measured. Therefore, it is very important to evaluate the degree of precision this relationship has in different tree species. Several non-linear functions have been used to relate trunk diameter and height of forestry stands (^{Juárez et al., 2006}; ^{Pece et al., 2006}; ^{Prodan, Peters, Cox, & Real, 1997}; ^{Vibrans, Moser, Oliveira, & Macaneiro, 2015}). The local equations used to describe CHD-Ht relationships are not complex like equations E1 to E10 in the research of ^{Trincado and Leal (2006)}, taken from several literature reports. However, different seasonal and forest stand conditions have led to a situation in which a single diameter-height equation would not fit all conditions, in such a way that an individual relationship must be fit according to differential development of individual forest land. However, this process requires huge amounts of data and is very costly (^{Schröeder & Álvarez-González, 2001}). Thus, these models must be of a local nature and should be fit individually to each tree stand growing under very particular environmental conditions of age, soils, climates, slope and exposures. What this suggests is that the extrapolation of general relationship is of limited value because the curves change according to a particular combination of such conditions e. g. low and high tree densities tend to produce respectively larger and smaller diameters for similar heights (^{Costa, Schroder, & Finger, 2016}; ^{López et al., 2003}).

Therefore, the objective of this research was to find a mathematical function that best relates diameter at chest height ( ≈ 1.3 m) to height in three species grown (Cinnamomum sp., Pinus chiapensis [Martínez] Andresen y Melia azedarach L.), in association in a subtropical climate of the State of Veracruz, Mexico.

Materials and methods

Results and discussion

Mean height was higher in M. azedarach, followed by P. chiapensis and Cinnamomum sp. The coefficient of variation values followed a reverse order, while the standard errors of the means were very low (0.06-0.07) due to the large number of observations (Table 2). Skewness was negative in the three species and absolute values of the ratios were larger than two, so height distributions had a peak significantly (P < 0.05) skewed to the right. Kurtosis was negative for Cinnamomum sp. and positive for the other species. In Cinnamomum sp. the ratio was < 2, so the distribution was platykurtic, showing a flat peak. In P. chiapensis and M. azedarach both ratio absolute values were much larger than two, indicating a narrow or leptokurtic peak.

Mean trunk diameter was smallest in Cinnamomum sp., intermediate in P. chiapensis and largest in M. azedarach. Variability was larger in wild avocado and smaller in Chiapas pine and piocho. Again, the standard error of the mean was small (0.09-0.12) due to the large N (Table 2). The three species showed negative skewness, thus diameter distribution had a peak to the left, but it differed significantly (P < 0.05) from the normal distribution only in Chiapas pine and piocho. Kurtosis was negative in wild avocado and positive in the other two species. Cinnamomum sp. and P. chiapensis had an almost normal peak (mesokurtic) while M. azedarach presented a narrow-peaked leptokurtic distribution.

The six non-linear models with the best fit for each tree species are presented in Table 3. In terms of Ra2 and AIC, the best model for wild avocado had two parameters, while in Chiapas pine and piocho the best models had three parameters. The fit was highest in wild avocado, intermediate in piocho, and lowest in Chiapas pine. Some models among the six best within each tree species resulted in exactly the same fit: models 9 and 20 in wild avocado; models 20 and 22 and 8 and 16 in Chiapas pine; and models 11 and 21, and 20, 22 and 12 in piocho.

Height-diameter scattering plots along with the best non-linear model for each tree species are presented in Figure 1. The shape of the predicted lines is quite similar, but the respective models were not, as the best models were: the two-parameter hyperbola (model 12) for wild avocado (^{Landsberg, 1977}); the three-parameter ^{Richards’ function (model 9) (Richards, 1959}; ^{Zeide, 1993}) for Chiapas white pine; and the three-parameter sigmoid function (model 11) of ^{Korf (1939)} for piocho (^{Stage, 1963}; ^{Zeide, 1989}).

Figure 1 Graphic representation of the best models to describe the trunk chest-height diameter to height relationship in three species grown in a commercial forest plantation in a semi-hot and humid climate of the State of Veracruz, Mexico. A) Wild avocado (Cinnamomum sp.) model 12; B) Chiapas white pine (Pinus chiapensis) model 9; C) Piocho (Melia azedarach) model 11. Open dots are data used to fit the models; closed dots were outliers excluded from the analyses. 

The analysis of variance showed that the model effect was highly significant (P = 0.0008) upon ranks. The model M19 presented a significantly (P ≤ 0.05) lower sum of ranks (a better fit on average) than models M01, M02, M03, M04 and M17. The sum of ranks of M09, M20 and M22 differed from that of M19 in 0.5, 0.5 and 1.0 rank units. The remaining models were statistically (P > 0.05) similar to M19, even though their sum of ranks were from 1.9 to 4.0 times larger (Table 4).

The best six models for each species differed very little among them with respect to their fit, as Ra2 and AICc values were very close among them (Table 3). Differences in Ra2 between the 1st and 6th model for wild avocado were only: 0.785 - 0.781 = 0.0041; and the respective difference in AICc was only: -1290.5 - (-1266.1) = -24.4. This is indeed a very low margin to declare superiority of one model over the remaining five (Table 3). Therefore, for predictive purposes only, any of the six best models can be used, unless the biological meaning of each parameter is of importance, if any, for the evaluator of the plantation.

Some models yielded the same Ra2 and AIC values. For example, model 11 (y=a*e-b*x-c); and model 21 (y=ea+b*x-c); the ‘a’ parameter for the former was 18.35 m, which predicts the maximum tree height for this set of data, while the ‘a’ parameter of the latter was 2.90; nevertheless, if the terms ‘b*(x^-c)’ are equaled to zero, e^2.90 = 18.17 m, not very different from the maximum height predicted by model 11. Therefore, most models with equal fit were just representing the same predicted line, expressed in a different algebraic way.

Model 12, the best for wild avocado, has a zero intercept, and thus fulfills the assumption that if no diameter exists, height must also be nonexistent, which is consistent with Pasteur’s Principle of Homogenesis v. gr. “…at zero mass, there is no potential for growth…” (^{Huxley, 1924}). This model provides an estimation of the asymptotic or maximum height that a species’ cohort should have reached had it had the optimum conditions to do so, in this case, 17.58 m (Table 3). It also produces a changing value of the slope [a*b/(b + x)²] as diameter changes, thus forming the curvature typically expected of the height-diameter relationship in several tree species.

The Richards’ function was the one that best fitted the H-D relationship in Chiapas white pine (model 9). This is a generalized logistic function which predicts maximum tree height at a = 10.14 m; it also has the “S” shaped form of the logistic, but it is inherently asymmetrical. The parameter c = 1.68, or point of inflection, indicates when the a/2 value (half the height asymptote) occurs on the diameter axis (^{Birch, 1999}), which in this case was (5.3 cm. It also predicts that at zero diameter height must be zero.

Korf’s function fit the data best in piocho. Here almost half the variation in height was explained by diameter. ^{Zeide (1993)} stated that this equation was first proposed in Czechoslovakia by ^{Korf (1939)} and he and subsequent researchers (^{Brewer, Burns, & Cao, 1985}; ^{Stage, 1963}) applied it to model height growth of forest stands with moderate success. The equation is quite suitable to describe diameter growth of a fixed number of trees. ^{Zeide (1989)} used the average growth of thousands of stem analyses of different species from different locations and showed that Korf’s model was more accurate than other models. The Richards, Weibull, Gompertz and logistic models had standard errors of the estimate that were 2.1, 2.3, 3.4 and 4.8 times larger than the model of Korf (^{Zeide, 1989}). Korf’s function predicted a maximum height of 18.25 m, a value that could only be reached at an unrealistic diameter of 1.29E+07 cm. Models 20 and 12 (Table 3) predicted asymptotes of 15.47 m and 15.08 m, respectively, which were closer to the largest height diameter measured in the field, which was 13.45 m (Table 2).

At any rate, the fit of the Richards’ function in Chiapas pine, in terms of the adjusted R², was lower than that shown by the hyperbola for Cinnamomum sp. and the Korf’s function for piocho (0.35 vs. 0.79 vs. 0.49; Table 3; Figure 1). ^{Barrett (2006)} modeled the height of 137 374 trees of 38 common species inventoried from 1988 to 2000 in 1 389 plots along the states of Washington, Oregon and California, USA, with the Richards’ function and found R² values ranging from 0.44 to 0.83 and that in general, conifers showed higher values than hardwoods but among the former, those with highly variable tree forms like Pinus contorta (Balf.) and P. occidentalis Swartz had lower R² values. This is contrary to our results. The most likely explanation for this discrepancy is that Chiapas pine and piocho, being non-native species, are outside their range of environmental conditions for optimum growth, and this leads to a high variability due to growth differences in response to individual sites where the trees are growing. It has been shown that for a given stand of trees of the same species with similar age and growing under similar environmental conditions, the non-linear relationship between diameter and height is affected by site quality and tree stand density e. g. young tree stands in good sites have a stronger slope, while trees in older stands in less favorable sites show a flatter slope (^{Prodan et al., 1997}). For our data, factors such as slope, exposure and soil differences remain to be tested. Several trees of each species should be measured, along with field traits like soil type, slope, exposure and others, and then tested to determine whether or not common conditions lead to a same best fit function or, alternatively, the functions do in fact change with field conditions.

In the practical sense, the mixed tree plantation took into account the different rates of growth and development of each tree species that will make them reach the adequate age and size for harvesting at different times, which at the end is related to the space for growth of the remnant trees v. gr. M. azedarach is the fastest growing, followed by P. chiapensis and Cinnamomum sp. in this order. In this way, if M. azedarach is harvested first, space is opened up and there is less competition for the other two species to grow; then at 16 to 22 years after planting P. chiapensis is harvested, leaving all space for Cinammomum sp. growth. ^{Domínguez (1986)} reported 18.3 m of height and 29.5 cm of chest-height-diameter at 12.5 years of age in M. azedarach, while P. chiapensis reached at 30 years of age, 25-30 m and 60-65 cm, respectively. On the other hand, empirical observations indicate that Cinnamomum sp., would require from 25 to 35 years of growth to have a height of 18 m and 80 cm of chest-height-diameter. These values would justify, initially, the type of tree species arrangement in the field available for the present research.

Another important aspect of the present evaluation is related to the scarce number of branches and leaves in M. azedarach and Cinnamomum sp. that would allow a greater amount of light to reach the ground, as compared to other more leaf-dense tree species. In this sense, the scarce shadow that M. azedarach generates as well as it being a winter-deciduous species makes it particularly adaptable as a tree component in agrosilvopastoral systems. An experiment conducted in the same property within a 1 200 trees·ha^-1 pure piocho stand compared the growth of four C₄ grasses during the rainy (August 2013) and windy (February 2014) seasons under sun and shade (^{Santiago-Hernández et al., 2015}). On average, shade resulted in a 107 % reduction in biomass from 4 851 to 2 342 kg·ha^-1, while rainy and sunny conditions produced eight times (8 326 kg·ha^-1) more biomass than windy, cold and shady conditions (1 071 kg·ha^-1). The establishment of silvopastoral systems using Cinnamomum sp. and M. azedarach appears to be a feasible alternative to improve the landscape and farmers’ income.

Conclusions

Our results agree in the general sense with the view that most biological processes like the growth of organisms is an imprecise phenomenon. The best models barely differed from others that fit the data almost equally well. Future research in this plantation must consider the present results and use the best functions to construct site indexes. Further investigation about the effects of slope, exposition and soil quality on the chosen best fit function should be pursued.