SPECIES-SPECIFIC BIOMASS EQUATIONS FOR SMALL-SIZE TREE SPECIES IN SECONDARY TROPICAL FORESTS [ECUACIONES ESPECÍFICAS DE BIOMASA PARA ÁRBOLES DE DIÁMETRO-PEQUEÑO EN BOSQUES TROPICALES SECUNDARIOS]

Colegio de Postgraduados, Km. 36.5 Carr. México-Texcoco, Montecillo, Texcoco, C.P. 56230, México; E-mail: puc.ramiro@colpos.mx; gangeles@colpos.mx; valdez@colpos.mx; vreyes@colpos.mx; perpdgo@colpos.mx Recursos Naturales, Centro de Investigación Científica de Yucatán (CICY), Calle 43 No. 130, Colonia Chuburná de Hidalgo, C. P. 97200, Mérida, Yucatán, México; Email: jmdupuy@cicy.mx Department of Geography, Rutgers University, 54 Joyce Kilmer Avenue, Piscataway, NJ 08854, United States; E-mail: laschnei@geography.rutgers.edu Instituto Nacional de Investigaciones Forestales, Agrícolas y Pecuarias, Campo Experimental Chetumal, Km. 25, Carretera Chetumal-Bacalar, C.P. 77930, Xul-ha, Quintana Roo, México; E-mail: xavier_garciacuevas@yahoo.com.mx *Corresponding author


INTRODUCTION
The interest of secondary tropical forests is increasing, not only for its extension or its role in biodiversity conservation, but also for its high resilience in biomass recovering from anthropogenic and natural disturbances (Chazdon et al., 2010;Memiaghe et al., 2016;Poorter et al., 2016).Very often, the biomass estimation in tropical forests have been based only with data of large trees (> 10 cm DBH) leaving smallsize trees (DBH < 10 cm) unaccounted, since the common assumption is that small-size tree biomass is less than 5% of large-size tree biomass (Lugo and Brown, 1992;Chave et al., 2003;Vincent et al., 2015).However, this assumption may not be applicable for all tropical forests because they are highly dynamic and their structure re the result of different factors (Vincent et al., 2015;Chazdon, 2003;Poorter et al., 2016).For instance, land use change leads to heterogeneous landscapes and canopy variations dominated by secondary forests that contain greater number of smallsize trees(here denoted as trees 1 cm ≤ DBH ≤ 10 cm) (Chave et al., 2003;Dupuy et al., 2012a;Memiaghe et al., 2016).Small-size trees are important component of the diversity of woody plants of secondary tropical forests and they can contribute significantly to the total stand biomass (Dupuy et al., 2012a;Memiaghe et al., 2016).
The secondary forests of the Yucatan peninsula in Mexico are dominated for small-size tress.In absolute values, the density of individuals > 5 cm DBH in forests of Campeche range from 1 000 to 3 422 trees per hectare in stands of 5 to 20 years-old, and if individuals > 2 DBH are included, the tree density may reach up to 4 000 trees per hectare (Schmook, 2010;Román-Dañobeytia et al., 2014).In relative values, the contribution of small-trees to the total density per unit area, has been reported in the order of 2.4% to 60% (Zamora et al., 2008;Gutiérrez-Báez et al., 2013).In other secondary tropical forests, for example in China, Kenia, Papua New Guinea, Costa Rica, and Panamá, small trees can reach 25% to 93.6% of the total tree density (Chave et al., 2003;Brandeis et al., 2006;Kuyah et al., 2012;Vincent et al., 2015;Memiaghe et al., 2016;He et al., 2018).Regarding tree biomass, small trees contribute between 3.6% up to 35.05% depending on the forest successional stage (Alief, 2012;Kuyah et al., 2012;Lima et al., 2012;Memiaghe et al., 2016).These observations are consistent with the idea that small-size trees can show a high variation in their contribution to the overall biomass.
The accurate biomass estimation of tropical forests depends on the quality of the allometric equations for structural tree components such as stem, branch and foliage, and total-tree biomass (Chave et al., 2005;Dong et al., 2014).Methods for estimating tree biomass could be classified into allometric equations at tree-level, structural component-level, and additive biomass equations (Parresol, 2001;Sanquetta et al., 2015b;Zhang et al., 2017;Bi et al., 2004).The treelevel method involves an independent modelling to estimate total-tree biomass directly, but lacks accurate information of stem, branches, and foliage biomass.The structural component-level, represents also an independent modelling and, total-tree biomass is obtained by summing up the biomass estimation by tree structural components.However, summing up the biomass estimation of structural components can diverge from total-tree biomass.Therefore, the additive method that consists in fitting the biomass data by structural components simultaneously, eliminates the inconsistency of the sum of the biomass by structural tree components on the same sampled trees.Thus, the total-tree biomass is the sum of the biomass of structural tree components.
Many studies mentioned that in tropical natural forests where hundreds of species coexist per hectare it is impractical to represent each specie with its own allometric biomass equation (Pappoe et al., 2010;Gibbs et al., 2007).While other authors indicate that specific-species can provide more accurate estimates of biomass and that help reflect better the real biomass in species-rich tropical ecosystems (Fehrmann and Kleinn, 2006;Henry et al., 2010;Kebede and Soromessa, 2018;Chaturvedi et al., 2012).Specificspecies equations are critical to evaluate the productivity of secondary tropical forests through tracking changes in the biomass stocks (Navar, 2009c;van Breugel et al., 2011;Cai et al., 2013;Cairns et al., 2003).Also, are key tools to understand the role of secondary tropical forests in the global carbon cycle through biomass estimations needed for accurate determination of carbon storage and fluxes (Litton and Boone, 2008;Navar, 2009c).In tropical regions, specific-species equations have been generated mainly for commercial species (Singh et al., 2011;Cai et al., 2013;Zhang et al., 2017).For natural forests they are scarce, but currently there are many works carried out in Costa Rica, Brazil, and India (Segura and Kanninen, 2005;Chaturvedi et al., 2012;Araújo et al., 2018).In Mexico, the species specific equations have been generated in temperate and warm tropical forests stands by Navar (2009c), Douterlungne et al. (2013), Cairns et al. (2003), andVargas-Larreta et al. (2017).Particularly, Cairns et al. (2003) generated 15 equations for species-specific in forests of southern Yucatán peninsula: nine for trees with DBH ≤ 10 cm, six for trees > 10 cm in DBH; as well they generated aa generic equation for tree species with DBH > 10 cm.The range of applicability of the equations developed by Cairns et al. (2003) for trees ≤ 10 cm in DBH is limited, due to the high species richness of small-size tree species that characterize the secondary tropical forests in the studied region.Besides, the equations were generated for old-growth stands species.Thus, the generic equation developed by Hughes et al. (1999) in the central-east, México, and Chave et al. (2003) equation in Panamá are widely used in the tropical forests of México to estimate biomass of trees with DBH ≤ 10 cm.However, the performance Hughes et al. (1999) and Chave et al. (2003) equations has not yet been evaluated with other equations generated for small-size trees in forests of the Yucatán peninsula.Therefore, the objectives of this study were (1) to develop specific-species equations under an additive system equations approach by tree structural components (stem, branch and foliage) (2) to develop allometric equations for total-tree biomass and (3) to compare the performance of the developed equations on biomass estimation to generalized equations developed for other tropical regions by Hughes et al. (1999) and Chave et al. (2003).The following hypotheses were tested: a) equations that include the total tree height (TH) and wood density (WD) as independent variables, besides DBH, provide better goodness of fit statistics than simpler equations (i.e. based on one or two predictors), since they include in their structure the effect of the allometric characteristics and wood properties that determine the species growth form; and b) specific equations developed in this study are more accurate to estimate total tree biomass at specie level, since it includes the effects biometric characteristics of local species (i.e. through DBH, TH and WD), than those generalized equations developed in other tropical regions.
The dominant ecosystem type is mid-stature and semievergreen tropical forest (Pennington and Sarukhán, 2005;Miranda and Hernández-Xolocotzi, 1963).The climate is tropical subhumid with mean annual rainfall between 948 and 1500 mm, most of which falls in the summer, while the dry months (april to march) typically have less than 60 mm of rainfall.The mean annual temperature is about 26° C with extremes of 36 °C to 38 °C in the driest months (García, 2003;Lawrence, 2005).There is a hurricane season from June to November, and the highest incidence occurs between August and October (Islebe et al., 2009;McGroddy et al., 2013).The topography is mostly flat with some places with slight inclination.Dominant soils are classified as gleysoles, vertic cambisols and vertic luvisols; they are thin and shallow, resulting in a slow water drainage and surface flooding's in the rainy season during storms or hurricanes (Ellis and Porter-Bolland, 2008).

Measurements of aboveground biomass
Biomass data was collected from harvesting 311 trees of 22 tree species of 1 cm ≤ DBH ≤ 10 cm (Table 1).The 22 species were selected based on the importance value index-IVI (Curtis and McIntosh, 1951) in a pre-assessment in the same study region.The index-IVI was determined for each species by summing the species relative frequency, relative density, and relative dominance Figure 1.Location of the study site in the southeast Yucatán peninsula.Based on the index-IVI values, between 12 to 18 trees for each specie were selected and harvested.Before felling the trees, DBH was measured with a diametric tape.After felling, TH (equal to the length of the stem) was measured with a long fiberglass tape.For biomass calculations; fresh weight for each component by tree (i.e.stem, branches, and foliage) were obtained with an electronic scale TORREY CRS-HD of 500 kg capacity (± 100 g).Three random samples ~ 100 g of stem, branches, and foliage were obtained with an electronic scale OHAUS Pionner tm 5 kg capacity (accuracy ± 0.1 g), respectively (Wang, 2006;Basuki et al., 2009;Gómez-García et al., 2013).The fresh samples of stem (disks of about 5 cm thick) were cut from the base, middle and upper part of the stem (Bastien-Henri et al., 2010;Picard et al., 2012).To determine WD (g.cm -3 ), samples were taken from each tree at 1.30 m from the base of the stem (Henry et al., 2010;Nam et al., 2016).
The WD was calculated as green volume/dry mass for each sample.The green volume was measured by the water displacement method (Chave et al., 2006).The stem, branches and foliage of trees ≤ 2.5 cm DBH were sent entirely at laboratory.All samples were ovendried at 70° C until they reached constant dry mass.Dry mass of stem, branches and foliage were calculated on the basis of the fresh mass with the average fresh to dry mass ratio of each component.The total tree aboveground biomass (kg) was calculated by adding up the total dry weight of three structural components of each tree.
The harvested tree species were classified in three WD classes (Table 1) according Sotomayor et al. (Sotomayor et al., 2010): low (≤ 0.40 g. cm -3 ), intermediate (0.41-0.60 g.cm -3 ) and high (≥ 0.61 g.cm - 3 ).These classes were used because the density values are associated with the wood quality.Likewise, WD is considered as an economic indicator -for the industryand a good wood descriptor to study the species ecological behavior (Chave et al., 2006;Sotomayor et al., 2010).

Fitting and evaluation of allometric equations
Total-tree biomass was modeled with independent variables of DBH, TH and WD from eight regression models (Table 2) previously tested in other studies for estimating tree-total biomass (Hughes et al., 1999;Ketterings et al., 2001;Cairns et al., 2003;Bi et al., 2004;Chave et al., 2005;Urquiza-Haas et al., 2007;Chave et al., 2014;Soriano-Luna et al., 2015): We used an independent fitting to estimates coefficient parameters of models applying the Newton's iterative method with nls function and fit technique Ordinary Least Squares (OLS) in R software (R Development Core Team).Weighting regressions were applied to the models to improve the homogeneity of variance and the goodness-fit-statistics (Parresol, 2001;Álvarez-González et al., 2007).
Table 2. Regression models tested to fit allometric biomass equations in secondary tropical forests of the southeast Yucatán peninsula.
No. Allometric model ), MSE = mean square error.We assumed that the error terms are distributed independent and identically distributed as ε~N(0, σ e 2 ).

Equation validation and simultaneous fitting
The goodness of fit-statistics: (i) root mean square error of the estimate (RMSE), (ii) the proportion of variance explained by model corrected by the number of parameters estimated (adjusted R 2 ) and (iii) were used to select the best regression models (Johnson and Omland, 2004;Ngomanda et al., 2014).We used the Akaike Information Criterion (AIC) only to select the best model from Table 2. Overall, the best model was the one which had low (RMSE) and (AIC), and high R 2 (Mugasha et al., 2016).
The best selected model was fitted simultaneously to generate equations by tree structural components.Then, equation for total-tree biomass is a function of the independent variables in the equations for structural component, including constraints on the parameters of the model, and can be expressed as: Where: AGB = above-ground biomass (kg); β = vector of regression parameters to be estimated, DBH = diameter at breast height (cm), TH = total tree height (m), WD = wood density (g.cm -3 ) for each specie.We assumed that the error terms are distributed independent and identically distributed as ε~N(0, σ e 2 ).The parameter vector was calculated using an iterative convergence process with Newton´s algorithm (SAS Institute Inc., 2011).The simultaneous fitting, was performed using the generalized least squares method called nonlinear seemingly unrelated regressions (NSUR) iteratively applying the ITSUR option of PROC MODEL in SAS (SAS Institute Inc., 2011;Sanquetta et al., 2015b).Once the models were fitted and the residuals were obtained, it is very common to detect heteroscedasticity in the residuals (Álvarez-González et al., 2007).In order, to correct this problem, we fitted models using weighted regression to improve homogeneity of variances and guarantee models additivity (Clutter et al., 1983;Parresol, 1999;Magalhães and Seifert, 2015).

Independent modelling
From the selected models, developed also specificspecies equations to estimate total-tree biomass.Likewise, we compared total-tree biomass observed against the total-tree biomass estimated with specificspecies equations and, generic equations generated by Hughes et al. (1999) (2) and Chave et al. (2003) (3).
Besides, the performance of each equations this study versus the generalized equations of Hughes et al. (1999) and Chave et al. (2003)  Where: AGB = aboveground biomass (kg), DBH = diameter at breast height (cm), TH = total tree height (m); exp = exponential function, ln = natural logarithmic function, WD = wood density (g.cm −3 ) and WD av = wood density mean of the evaluated plot (0.54 g. cm −3 ).
We selected the Hughes et al. (1999) and Chave et al. (2003) generalized equations to evaluate their performance in comparison with equations developed in this study.Hughes et al. (1999) and Chave et al. (2003) equations were generated for a mixture tree species with DBH ≤ 10 cm, which is the same size range trees used in this study.Hughes et al. (1999) equation was developed for a tropical forest in centraleast, México, and its application is based only in DBH as biomass predictor variable.Chave et al. (2003) equation, also was developed to estimate biomass of small trees in tropical forests of Panamá, it is a reparametrization of Hughes et al. (1999) model, and it includes the WD as a second independent variable, in addition to DBH.We anticipated the addition of WD as biomass predictor to be beneficial for models performance and accuracy.However, let us believe that Hughes et al. (1999) and Chave et al. (2003) equations has a disadvantage, that its application is limited outside the forests of central-east, México and Panamá owing variables used in both equations are linked or express the biometric characteristics of forest species where the two equations were developed.Specific-species equations developed in this study, Hughes et al. (1999), and Chave et al. (2003) equations relative mean error (RME %) was calculated as validation criteria of performance of the allometric models (Djomo et al., 2010;Chave et al., 2014;Goodman et al., 2014) (4): Where: AGB ̂pred and AGB obs are the sum of the predicted and observed biomass for each tree species.
The RME (%) were computed using leave-one-out cross-validation for specific-species equations developed in this study, and Hughes et al. (1999) and Chave et al. (2003)) equations (Efron and Tibshirani, 1993).The principle of the leave-one-out method was to fit the model from  − 1 observations and calculate the RME (%) to the observation leave out (data not included in the training) or data validation (Efron and Tibshirani, 1993;Sileshi, 2014).The leave-one-out cross-validation was carried out for each species (22 total).Negative and positive values of the RME (%), indicate underestimation and overestimation of the biomass for a set of trees (Kuyah et al., 2012;Goodman et al., 2014).We used Kruskal-Wallis test at 95% confidence intervals with kruskal.testfunction of the "stats" package in R (R Development Core Team) to analyze differences in total-tree biomass estimate, and RME (%) values between equations.Furthermore, accuracy of the biomass equations of this study and, Hughes et al. (1999) and Chave et al. (2003) equations was analyzed with a linear regression between the predicted and observed biomass values (without intercept) using the lm function in R (R Development Core Team).If the models correctly fit the data then the slope of the estimated coefficient should be around one; values that are not around one indicates lack of fit of the model (Piñeiro et al., 2008;Sileshi, 2014).

Biomass fitted equations
Results of goodness of fit statistics as modeling performance criteria for total-tree biomass of all data sets (22 tree species) are presented in Table 3.All allometric models fit well to total-tree biomass, and most of them explained more than 90% of the observed biomass variance.Considering the smaller values of RMSE, AIC, and higher R 2 , the model I and II showed the best fit compared to the other models.The estimated coefficients parameters were significantly different from zero in all species-specific equations, which suggest that among the variable combined of model I (DBH 2 H) and II (WDDBH 2 TH) exist a strong relationship with total-tree biomass.

Biomass equations by structural components
The weighting regression 1 Dn 2 H was adequate to improve the homogeneity of variance and goodness of fit statistics of the two models.Simultaneous fitting by tree structural component biomass was carried out using the model I structure (Table 4), because it showed the best goodness of fit statistics since the parameter estimates into the system of equations for each specie converged iteratively using the option ITSUR.According to the RMSE an R 2 values, the allometric equations were more accurate for stem and total-tree biomass across 22 species than branch and foliage biomass.The observed biomass variance explained by fitting the model I by tree structural components was: stem, between 64% and 99%; branch, 24% to 95%; foliage, 35% up to 94%, respectively.While, total-tree biomass equations explained 60% to 99% of the total observed biomass variance.Specifically, model I explained less than 70% the observed biomass variance for three components: stem biomass of Bursera simaruba (64%); branch biomass in B. simaruba (24%) and Piscidia piscipula (66%); and, foliage biomass in Lonchocarpus xuul (35%), Zuelania guidonia (57%) and Psidium sartorianum (58%).), MSE = mean square error.We assume that the error terms are distributed independent and identically distributed as ε~N(0, σ e 2 ).1.49 0.95 DBH = diameter at breast height (cm); TH = total tree height (m), AGB ̂s =stem biomass estimate (kg), AGB ̂b =branch biomass estimate (kg tree −1 ), AGB ̂f =foliage biomass estimate (kg ), AGB ̂total−tree = AGB ̂s + AGB ̂b + AGB ̂f (kg), RMSE = root mean square error of the estimate, and Adjusted R 2 = proportion of variance explained by model corrected by the number of parameter estimates.

Species-specific biomass equations
The coefficients of the parameter and goodness of fit statistics of the species-specific equations to estimate total-tree biomass are showed in Table 5.We developed the species-specific equations using the structure of model II, because it showed the best goodness of fit statistics to estimate total-tree biomass and the parameter estimates for each specie converged iteratively using the independent fitting approach.Thus, the combined predicting variable (WDDBH 2 HT) captured most part of trees biomass.
Additionally, the effect of coefficients (β ̂1) that represent the influence of the three variables (WDDBH 2 HT) on the biomass were all significantly different from zero at α = 0.5.The R 2 values showed that the explained observed biomass variance by fitted models were between 94% and 99%.Total-tree biomass estimation was less accurate in B. simaruba, and Cecropia peltata, judged by the RMSE and R 2 values.

Comparison observed versus estimated total-tree biomass
There was a significant linear relationship between predicted biomass with species-specific equations and observed biomass in most individual species (Figure 2).The correlation values () were between 0.96 and 0.99 for all species.Three species showed a slightly underestimated biomass owing to the slope value of linear regression ( ̂1) being slightly above to one: L. xuul ( = 1.0119),Simarouba glauca ( = 1.01448) and C. peltata ( = 1.1577) (Figure 2).Across all species, the slope value obtained with Hughes et al. (1999) equation was between  = 1.206581 and  = 2.53534 (Figure 2).With Chave et al. (2003) equation the coefficient was between  = 1.17703 and  = 1.99099 (Figure 2).The results indicate that these two equations underestimated observed biomass observed in the majority of tree species, in particular for species with high and intermediate WD.We obtained statistical differences (Kruskal − Wallis test; χ 2 = 9.78, d. f = 2, p ≤ 0.007) on the median of estimated biomass among equations of this study and those developed by Hughes et al. (1999) and Chave et al. (2003) .
Figure 2. Observed total-tree biomass versus estimated total-tree biomass with species-specific equations fitted in this study, and Hughes et al. (1999) and Chave et al. (2003) equations.The orange solid line represents the 1:1 ratio between the biomass values.The red, green and blue solid line represents the linear regression between observed and predicted biomass.

Comparison of biomass equations performance
A comparison of the relative mean error (%) of the estimated biomass with the species-specific equations of this study, Hughes et al. (1999), andChave et al. (2003) equations is showed in Table 6.Specificspecies equations developed in this study overestimated between 0.36% and 2.97% on average the total-tree biomass, which was registered mainly for Dipholis salicifolia and Lysiloma latisiliquum.While they underestimated total tree-biomass between 0.14% and 9.41% in Chrysophyllum mexicanum and C. peltata.Chave et al. (2003) equation underestimated biomass between 7.38% and 48.46% on average.These values were recorded for L. latisiliquum and Licaria campechiana, respectively.Besides, Chave et al. (2003) equation does not overestimate total tree biomass of any species.When using Hughes et al. (1999) equation, an overestimation total tree biomass was observed between 33.04% and 44.22% on average, which corresponded to B. simaruba and C. peltata.While it underestimated total biomass between 12.10 and 59.82%, and it was observed in S. glauca, Pouteria unilocualris and Psidium sartorianum, respectively.In most of the species, there were significant statistical differences (Kruskal − Wallis test; χ 2 = 85.7, d. f = 2, p ≤ 0.0001) in the relative mean error of estimated biomass among equations of this study and, Hughes et al. (1999) and Chave et al. (2003) equations.Table 6.Comparison of the relative mean error (%) of the estimated individual total-tree biomass between the speciesspecific equations developed in this study and those developed by Hughes et al. (1999) and Chave et al. (2003).

Species name
Hughes

DISCUSSION
We developed species-specific equations for estimating biomass by structural components (i.e.stem, branch and foliage) and total-tree biomass for small-size trees in secondary tropical forests of the southeast Yucatán peninsula.When we modeled all biomass dataset, the model I and II (Table 2 and 3) presented the best goodness-fit-statistics.The addition of TH and WD in the models as predictors were key variables to improve the models fit.In much of the previous studies, only DBH has been widely used as a predictor variable to fit biomass equations, with which reliable biomass estimates have been obtained in tropical forests (Basuki et al., 2009;Douterlungne et al., 2013;Vargas-Larreta et al., 2017).Other studies use only DBH and WD (Svob et al., 2014;Chaturvedi et al., 2012;Nam et al., 2016), but not TH because of the difficulties that represent to precise measure it in the field (Segura and Kanninen, 2005;Djomo et al., 2010;Hunter et al., 2013).However, our results are consistent con other studies that recognize the need to integrate both TH and WD in models, since they substantially reduce the error in model fitting, and increase the accuracy in biomass estimation (Feldpausch et al., 2012;Chave et al., 2014;Mugasha et al., 2016).

Biomass equations by structural components
Simultaneous fitting of biomass equations by tree structural components were satisfactory with the model I (Table 2), since the total-tree biomass was equal to the sum of the biomass of structural components.Our findings on the additivity of total biomass for individual tree species was consistent with previous studies across the world in temperate forests (Bi et al., 2004;Soriano-Luna et al., 2015;Sanquetta et al., 2015b;Vargas-Larreta et al., 2017;Zhang et al., 2017).In tropical forests, it is not yet frequent studies that fit biomass equations with the simultaneous fitting approach.Besides, other studies have shown that the application of the SUR technique in simultaneous fitting is a successful methodology, because it takes into account the inherent correlations among biomass components to the total-tree biomass (Parresol, 2001;Bi et al., 2010;Dong et al., 2014).While, an independent fitting at component level the sum of components of biomass can generate inconsistent results in total tree biomass, since it does not guarantee the principle of additivity (Sanquetta et al., 2015b;Zhang et al., 2017).

Performance of biomass equations
In tropical forests there are few studies that have compared the equations performance developed for trees ≤ 10 cm DBH.We compare the performance of species-specific equations versus generalized equations by Hughes et al. (1999) and Chave et al. (2003).Across 22 tree species, species-specific equations developed in this study underestimated ~ 9.5 % on mean total-tree biomass, and overestimated in ~ 3%.Species such as B. simaruba showed the higher relative error (Table 6).We observed that the branches of B. simaruba were very widespread and of irregular size.Thus, we believe that this pattern of biomass allocation influenced the accuracy of the fit of branch equation of this species (Table 4).It was revealed that branch equations are less accurate since biomass is naturally more variable than the other components (Sanquetta et al., 2015b).For instance, many tropical species and broadleaf species in temperate forests, tend to extend their branches to reach the largest amount of photosynthetic light for tree growth and development (Sanquetta et al., 2015b;Ngomanda et al., 2014;Zhang et al., 2017).This implies that the architectural type of the plant can affect the variability of the branch biomass (Ngomanda et al., 2014).
The performance Hughes et al. (1999) equation was inconsistent across the 22 species evaluated in this study.For instance, the error of biomass overestimation for species with low WD (B.simaruba and C. peltata) was ~ 33% up to 44%.While the error of biomass underestimation in species with high WD (mainly P. unilocularis and P. sartorianum) was ~ 12% to 60%.Hughes et al. (1999) equation did not express with accuracy the biomass of species with wood densities different of the species harvested in stands used in this study.We also observed in our study that B. simaruba and C. peltata showed higher root mean square error (0.079 and 0.063) than the other species.Our results are consistent with studies reported in dry and wet tropical forests of India and Tanzania (Kebede and Soromessa, 2018;Mugasha et al., 2016), which indicated that WD can affect the models fit since it differs among tree genus and species, such as the species with low and high WD of this study.
When we applied the equation developed by Chave et al. (2003) to our data, the biomass estimation at species level improved slightly compared to the Hughes et al. (1999) equation, possibly due to inclusion the WD in the equation structure.The performance of Chave et al. (2003) equation was variable in the underestimation biomass of all the species (between 7% and 48.5%), regardless of the number of sampled trees per species.The higher error was recorded in species with high WD (0.61-0.80 g.cm -3 ).Svob et al. (2014) pointed out that when the error of the models differs between tree species, independently of the sample size, the equation should not be applied outside the area where they were generated.The high error levels in biomass estimations registered with Hughes et al. (1999) and Chave et al. (2003) equations, help us to confirm that the use of equations generated for regions different than the site where they are intended to be applied is a significant source of uncertainty in estimating local biomasscarbon stocks (Sileshi, 2014).

Sources of error in biomass equations
The larger bias in biomass estimates with Hughes et al. (1999) equation, might be associated to the use of DBH as the only predictor variable, the sample size that was used (66 trees), and the absence of TH and WD as parameters that are linked to forest structural and growth of the species (Svob et al., 2014).. Particularly, most biomass equations based only on DBH as biomass predictor, could represent a significant bias for total-tree biomass estimates (Chaturvedi et al., 2012), since only DBH is insufficient to explain the whole variation of total-tree biomass (Navar, 2009c;Feldpausch et al., 2012).Therefore, if both TH, WD, and crown diameter are considered in model as predictor variables, there may be a stronger relationships total-tree biomass (Navar, 2009c;Ngomanda et al., 2014;Duncanson et al., 2015).For instance, in Kenia and México in tropical forests the species-specific equations that relate biomass to DBH only may reached biases between 4.6% to 18% (Kuyah et al., 2012;Douterlungne et al., 2013).While generic models, the biomass could be overestimated up to 52% in global scale forests (Djomo et al., 2010;Chaturvedi et al., 2012;Chave et al., 2014).In some tree species of forests of Mexico, India, China, and Thailand, the RMSE of models fit with only DBH vary between 0.55 up to 0.65 (Chaturvedi et al., 2012;Douterlungne et al., 2013;Cai et al., 2013;Ounban et al., 2016).This range values were higher compared with RMSE of 0.004 to 0.111 found in this study.We confirmed that TH and WD in our model I improved the RMSE compared to other models tested.The combined variable (WDDBHTH) in model I, also has been applied in tropical forests of Africa, Asia, Panamá, and Brazil (van Breugel et al., 2011;Goussanou et al., 2016;Nam et al., 2016;Lima et al., 2012).Estimates of total-tree biomass at local scale have been reliable.This type of models more accurately estimates local biomass since they are parameterized with local data of TH and WD, which are strongly correlated with the total-tree biomass.For instance, in African and Asian forests, local equations underestimated 10% and overestimated total biomass up to 23.9%.But, if applied pan-tropical equations that include combined variable for estimating biomass at local scale, the bias in biomass estimation can be up to 62% (van Breugel et al., 2011;Bastin et al., 2015;Goussanou et al., 2016).With our models, we obtained ~ 3% on mean error of the biomass underestimation and ~ 9.5 biomass overestimation.
Regarding  et al. (1999) might be affected by sample size and the lack of TH and WD as predictor variables , which varies among forests at regional scale (Svob et al., 2014;Ketterings et al., 2001;Chave et al., 2006).Likewise, in dry and wet tropical forest of Costa Rica, Svob et al. (2014) observed than the performance of biomass models depends of the sample size, since the uncertainty of the biomass estimation increase when number of sampled trees decreased.Chave et al. (2003) included in their equation a constant value of 0.54 g.cm -3 , that corresponds to the mean WD obtained from 123 species in tropical forest of Panamá.However, it appears that the wood densities by species were not obtained entirely in the region where the equation was developed.In contrast, wood densities that we used in our study were based on tree samples that we took in the field.The equations developed in this study showed better performance in biomass estimation that those developed by Hughes et al. (1999) and Chave et al. (2003).Similar results have been found elsewhere when a generic equation developed from a different region is applied at local level.For instance, Ketterings et al. (2001) generated equations for specific sites with trees of 5 to 50 cm in DBH (29 trees total) in secondary forests of Sumatra; further, they contrasted the performance of their equations with those generated at global scale from data collected in a wide range of tropical climatic conditions and tropical vegetation types by Brown (1997).The Ketterings et al. (2001) equations , which included WD in addition to DBH, reduced the estimation error of total biomass by 36-51%, but the estimates were significantly higher than those observed when they applied Brown (1997) equation.In other studies, such as the one carried out in Brazilian forests, 10.6% and 14.8% mean estimation error were observed with the pan-tropical equations of Brown (1997) and Chave et al. (2005), whereas the local models showed 5.63% mean estimation error (Lima et al., 2012).In southeast Asian forests, 19.8% on mean error was obtained locally, but when regional and global scale equations were used the mean error was from 31.2% to 38.4% (Nam et al., 2016).
The higher relative estimation error showed by biomass equations may be due to several factors: 1) when the equations are fitted with a relatively small sample size, the coefficients are not adequate for other sites where may be higher tree densities and species richness, which may not be similar to the range of treediameters used in the equation fit process (Ketterings et al., 2001;van Breugel et al., 2011;Svob et al., 2014); 2) the use of only DBH as predictor variable may not be sufficient to explain the relationship with total tree biomass, because it does not express all the functional and ecological plant features (Feldpausch et al., 2012;Goussanou et al., 2016); 3) the absence of the effect of TH and WD in the equations, can decrease the prediction capability of the developed equations (Feldpausch et al., 2012;Mugasha et al., 2016;Valbuena et al., 2016).It is widely known, the TH and WD vary among tropical regions due to species composition and site characteristics (i.e.soil type, forest age, and precipitation), in which both variables can modify the equations coefficients (Chave et al., 2014;Valbuena et al., 2016) and may not be appropriate for sites outside where the equations were developed (Ketterings et al., 2001;Baker et al., 2004;Cai et al., 2013;Mugasha et al., 2016).Therefore, considering TH and WD as explanatory variables in biomass equations is important to account the biomass variation triggered by environment factors (Feldpausch et al., 2012;Mugasha et al., 2016).In this study we did not use mixed-effects models to analyze the effect of species and sites (covariates).However, we reiterate that the fitted equations implicitly take into account in their structure, through WDDBH 2 HT , information that accounts the variability effect of local species and site environmental conditions.Consequently, we consider that the equations are efficient and statistically reliable for biomass estimate of small trees in study region.On the other hand, these equations are parsimonious and are applicable for inventory data in other similar forest types owing the inclusion of WD and HT in their structure, and can lead to an improvement in biomass estimation in tropical secondary forests with high species richness.

CONCLUSIONS
We generated species-specific biomass equations for small-size trees of 22 species of secondary tropical forests of the Yucatán peninsula.The biomass equations were fitted by structural components such as stem, branch and foliage, and total-tree biomass. .Our results supported the hypothesis that using TH and WD, besides DBH, as predictor variables in the equations can appreciably reduce the error in the goodness of fit statistics, and better explain total-tree biomass variation.The predicting ability of the species-specific equations here developed was higher in the estimation of total-tree biomass at species level than generic equations developed for other tropical regions.Therefore, we confirmed the hypothesis that local species-specific equations are statistically more precise for biomass estimations at individual-tree level, since they account for the allometry variability of tree species of the site.The individual equations are adequate to improve biomass stocks estimations, while, generic equations may deliver higher uncertainties which needs to be considered when they are applied at local scale.Our equations could be used to reduce the uncertainty on biomass-carbon stocks estimations of tropical secondary forests that contain high densities of small-size tree species, such as those commonly present across the Yucatán peninsula.They may be useful for carbon estimation within climate change contexts to evaluate the role of tropical forests in carbon removals and emissions.
Pecuarias (INIFAP), Campo Experimental Chetumal, for allowing us the use of their facilities to process biomass samples.

Table 1 .
Tree species sampled to fit allometric biomass equations in secondary tropical forests of the southeast Yucatán peninsula.
was evaluated:

Table 3 .
Goodness of fit statistics for the allometric models fit to 22 tree species ≤ 10 cm diameter of secondary tropical forests of southeast Yucatan peninsula.

Table 4 .
Biomass equations and goodness of fit statistics by tree structural components for mall-size trees (≤ 10 cm DBH) in secondary tropical forests of southeast Yucatán peninsula.

Table 5 .
Biomass equations and goodness of fit statistics for species-specific equations for trees ≤ 10 cm diameter of secondary tropical forests of southeast Yucatan peninsula.
van Breugel et al. (2011)ize on biomass models,van Breugel et al. (2011)compared the performance of models with different sample sizes.The authors fitted two generic local models using 244 trees of 26 species in Panamá secondary forests; using for one model only the DBH as independent variable, and DBH and WD for the second model, respectively.When these authors used 80% (195 trees) and 20% (49 trees) of the total sampled trees, the relative mean error of models increased from 4% to 21%.They concluded that to develop an equation that include only DBH, it is necessary a larger sampling size than a model that includes DBH and WD, since the model parameters are systematically sensitive to small sample size.In that sense, the performance of the generic equation of Hughes