#### 3.2. Goodness of Fit Tests

The Weibull density function projected diameter distributions compatible with the observed pine diameter distributions, according to the χ

^{2} and K-S tests. MV2 and MV3 consistently accepted the highest percentage of null hypotheses (76.5%), in contrast to MZM and MDS (34.9% and 44.6%, respectively). MV2 and MV3 procedures also accepted the largest percentage of null hypotheses, according to the K-S goodness of fit test (95.1%). The MDM and MPP techniques, however, recorded the least goodness of fit with 33.0% and 58.1% of null hypotheses accepted, respectively (

Figure 2).

Using the χ

^{2} model, the approaches MV2 and MV3 had the largest percentage of accepted null hypotheses (60.1) for oak trees. The MCM and MDS techniques recorded the worst goodness of fit with only 25.5% and 29.4% of accepted null hypotheses. Using K-S, the MCM and MV2 methods recorded the largest percentages of accepted null hypotheses (92.9% and 90.4%, respectively). On the other hand, MDS and MRZ had the worst goodness of fit with only 38.8% and 43.8% of null hypotheses accepted, respectively (

Figure 2).

**Figure 2.**
Goodness of fit tests χ^{2} and K-S conducted on nine different techniques of parameter estimation of the Weibull density function for 587 forest stands of fitting (**a**, **b**) and 250 forest stands of validation (**c**, **d**) parameters.

**Figure 2.**
Goodness of fit tests χ^{2} and K-S conducted on nine different techniques of parameter estimation of the Weibull density function for 587 forest stands of fitting (**a**, **b**) and 250 forest stands of validation (**c**, **d**) parameters.

In general, pines have smoother diameter distributions than oaks. As the Weibull density function failed to fit well the remaining forest stands (23.5% and 39.9%), it is recommended to apply other probabilistic distribution functions, such as the Johnson SB density function. Borders and Patterson [

44], Cao and Baldwin [

45], Kangas

et al. [

46], Návar-Cháidez [

37], Parresol

et al. [

23] and Návar-Cháidez and Dominguez-Calleros [

47] described other non-distributional approaches to predict the diameter distributions of forest stands. For quite a few, managed multi-cohort forests the diameter distributions were better described by a bimodal distribution function compared to the unimodal one; as it was the case for cedar forests in Morocco [

48]. However, they were so few that I decided to go on with the unimodal Weibull distribution model.

#### 3.3. Parameter Variance and Bias

For pine trees, methods MV2, MV3, MNP and MPP had the smallest parameter variances for α, β and ε (0.001, 0.001 on average, 0.067, and 0.013, respectively), unlike the approach MCM, which recorded the largest variance for all three parameters. For oaks, parameter estimators α and β, when calculated by techniques MV2, MV3 and MNP, had the smallest variances (0.007 and 0.328 on average, respectively). The MPP method showed the smallest variance for ε (0.071). All three parameters exhibited the largest variance when estimated by the MCM method (

Table 3).

**Table 3.**
Efficiency and consistency of parameter estimators for α, β and ε calculated by nine methods for pines and oaks in mixed forest stands of Durango, Mexico.

**Table 3.**
Efficiency and consistency of parameter estimators for α, β and ε calculated by nine methods for pines and oaks in mixed forest stands of Durango, Mexico.
Method | Weibull Distribution Parameters |
---|

α * | β ^{†} | ε ^{‡} |
---|

BP ^{§} | A ^{||} | S² ^{¶} | BP | A | S² | BP | A | S² |
---|

| | | | Pines | | | |

MNP | 0.027 | 1.24 | 0.009 | 0.085 | 25.52 | 0.067 | −0.023 | 13.89 | 0.048 |

MPP | −0.085 | 1.23 | 0.004 | −0.022 | 11.48 | 0.086 | 0.015 | 14.01 | 0.013 |

MCM | 0.234 | 1.76 | 0.696 | −1.641 | 25.77 | 14.97 | −4.216 | 11.17 | 40.220 |

MV2 | 0.035 | 2.86 | 0.001 | −0.053 | 28.23 | 0.084 | ** | 12.50 | ** |

MRZ | 0.227 | 1.91 | 0.010 | 0.300 | 25.80 | 0.233 | −0.160 | 10.59 | 0.127 |

MDS | 0.050 | 2.34 | 0.009 | 0.358 | 22.90 | 0.378 | −0.400 | 4.16 | 0.476 |

MRM | 0.024 | 1.27 | 0.008 | 0.065 | 11.62 | 0.237 | −0.017 | 13.93 | 0.059 |

MZM | −0.059 | 0.98 | 0.018 | −0.085 | 25.80 | 0.098 | 0.025 | 14.53 | 0.016 |

MV3 | −0.057 | 1.18 | 0.058 | −0.882 | 12.04 | 13.19 | 0.322 | 13.33 | 1.559 |

| | | | Oaks | | | |

MNP | 0.013 | 0.97 | 0.008 | 0.168 | 25.84 | 0.328 | 0.075 | 13.82 | 0.221 |

MPP | 0.027 | 0.92 | 0.007 | 0.401 | 11.37 | 0.665 | −0.128 | 14.24 | 0.071 |

MCM | 1.525 | 2.39 | 5.966 | −1.479 | 26.64 | 18.71 | −9.453 | 5.25 | 27.969 |

MV2 | −0.006 | 2.17 | 0.007 | 0.088 | 29.50 | 3.497 | ** | 13.00 | ** |

MRZ | 0.217 | 1.55 | 0.030 | 0.250 | 26.25 | 0.736 | −0.173 | 9.77 | 0.650 |

MDS | 0.091 | 1.86 | 0.007 | 1.060 | 25.39 | 0.909 | −0.770 | 3.57 | 0.610 |

MRM | 0.016 | 0.97 | 0.008 | 0.107 | 12.01 | 0.975 | 0.037 | 13.81 | 0.241 |

MZM | 0.041 | 0.75 | 0.010 | 0.432 | 26.38 | 0.475 | −0.009 | 14.56 | 0.285 |

MV3 | −0.091 | 0.86 | 0.50 | −0.231 | 10.47 | 31.94 | 0.321 | 14.31 | 10.74 |

For pines, the location parameter α was least biased when parameters were estimated by MRM, MNP, MV2, and MV3 (0.024, 0.027, 0.035, and 0.035, respectively). The parameter β was also least biased when estimated by MPP, MV2, MV3, and MRM (−0.022, −0.053, 0.021, and 0.065, respectively). Finally, the smallest bias for ε was recorded when MPP and MRM were used to estimate parameters (

Table 3). For oak diameter distributions, α and β were least biased when MV2 was used to estimate parameters (−0.006 and 0.088, respectively), and ε was least biased when MZM (−0.009) estimated parameters. MCM and MDS recorded the largest parameter bias for all three estimators (

Table 3).

The procedure of maximum likelihood of two and three parameters consistently yielded compatible diameter distributions similar to those measured for pines and oaks. The assumption that ε = 0 seems to work well because these forests are under management, and contiguous forest openings allow the establishment of regeneration, which in turn allows the contiguous presence of the smallest inventoried diameter classes. Hence, I stress the appropriateness of either the two or three- parameter maximum likelihood techniques of parameter estimation to simulate the diameter distribution of these forests. The former technique of parameter estimation has also been recommended to probabilistically model the diameter classes of

Q. robur [

49] and

P. elliottii [

30], as well as the diameter classes of pine, oak and juniper trees of native mixed and multiaged forests [

26]. This method is validated by the fact that it produces estimators, which meet the statistical requirements of efficiency, consistency, unbiasedeness, and small variance. Haan [

24] pointed out that this procedure uses all the information available when calculating parameter estimators, and that it converges well into the constant value with a small number of observations. Nanag [

50] noted that the maximum likelihood, percentile, and moment procedures of parameter estimation produce compatible results. Other investigators recommended the percentile technique because of the ease with which it can be used to estimate parameters [

51,

52].

#### 3.4. Parameter Prediction and Recovery

Using the MV2 procedure of parameter estimation, α and β as well as Xp, Std, and Sk were evaluated by the following equations (

Table 4).

**Table 4.**
Weibull parameter prediction and recovery equations for pine and oaks growing in Mexico’s northern mixed multiaged temperate forests.

**Table 4.**
Weibull parameter prediction and recovery equations for pine and oaks growing in Mexico’s northern mixed multiaged temperate forests.
Group of Species | Parameter | Empirical Equation | n | r^{2} | Sx |
---|

Pinus spp. | α | 0.91Dm^{13.86}Dq^{−12.31}N^{−0.49}BA^{0.41} | 587 | 0.52 | 0.20 |

β | 2.9 + 2.2Dm − 0.01N − 1.2Dq + 0.13BA + 0.005Cc + 0.033H | 587 | 0.93 | 0.89 |

Xp | 98.5N^{−0.46}BA^{0.46} | 587 | 0.96 | 0.55 |

Std | 1.36 + 0.29Xp − 0.008N + 0.13BA | 587 | 0.68 | 1.10 |

Sk | 0.00000021Std^{2.62}N^{3.14}BA^{−2.93} | 587 | 0.42 | 0.51 |

Quercus spp. | α | 0.30β^{0.99}Dm^{4.18}Dq^{−4.48}IDR^{−0.068} | 587 | 0.51 | 0.21 |

β | 12.76 + 1.77Dm − 0.035N − 1.13Dq + 0.56BA | 587 | 0.84 | 1.39 |

Xp | 92.8N^{−0.43}BA^{0.43}Cc^{−0.031} | 587 | 0.94 | 0.76 |

Std | 799902177Xp^{−3.5}N^{−2.5}BA^{2.5}Cc^{0.061} | 587 | 0.88 | 1.61 |

Sk | 0.00004Std^{1.9}N^{1.9}BA^{−1.8}Cc^{0.15} | 587 | 0.50 | 0.57 |

When predicting parameters from stand attributes, the validation procedure for the independent data set of 250 stands indicated that 51.3% and 74.7% of the population accepted the null hypothesis when using the χ

^{2} and K-S tests, respectively. When using the moment recovery approach, 49.8% and 78.4% of the population fit the two-parameter Weibull distribution when using the χ

^{2} and K-S tests, respectively. For oaks, the moment prediction approach and the recovering of parameters by MNP accepted 37.5% and 73.7% of the population, as tested by the χ

^{2} and K-S statistics, respectively. The parameter prediction approach was accepted by only 43.7% and 66.7% of the population, as tested by the χ

^{2} and K-S goodness-of-fit tests, respectively. When averaging the results of χ

^{2} and K-S tests, the parameter recovery approach fitted diameter distributions better than the parameter prediction approach, which is consistent with the findings of Hyink [

34] and Gove and Patil [

53].

#### 3.5. Sensitivity Analysis

The prediction equation of the diameter distribution of pines was most sensitive to the standard error in α because the percentage of accepted null hypotheses was reduced on average to 85% for the χ

^{2} and K-S goodness-of-fit tests. Changes in the acceptance of null hypotheses were hardly noticed when the standard error was added to the equation to predict β. The percentage of accepted null hypotheses notoriously shifted to 76.6% when the standard errors of each parameter were added (

Table 5). The sensitivity analysis reflects the need to predict and recover α with the greatest precision. Návar

et al. [

38] demonstrated that growth models based on predicting the Weibull distribution parameters were most sensitive to changes in α values as well. Návar-Cháidez [

37] developed a simple regression equation to predict α with the skew coefficient, Sk, consistent with the mathematical theory.

For the parameter recovery approach, the equations used to predict the moments indicated that Sk must be estimated with the greatest precision because the percentage of accepted null hypotheses was highly sensitive to this statistic. The percentage of accepted null hypotheses was reduced on average by 20%. The contribution of the error to the standard deviation and the mean was small (less than 5% of change). However, when all standard errors were incorporated into the equation, the percentage of accepted null hypotheses was reduced by 28% (

Table 5).

**Table 5.**
Sensitivity analysis of equations for predicting and recovering the Weibull distribution parameters, and the goodness-of-fit test for pine and oak stands of Durango, Mexico.

**Table 5.**
Sensitivity analysis of equations for predicting and recovering the Weibull distribution parameters, and the goodness-of-fit test for pine and oak stands of Durango, Mexico.
Parameter | Ho Accepted (χ^{2}) | Ho Accepted (K-S) |
---|

Prediction Approach | Pine | Oak | Pine | Oak |

No change MV2 | 51.3 | 43.7 | 74.7 | 66.7 |

α ± EES | 32.5 | 36.7 | 71.9 | 65.2 |

β ± EES | 49.8 | 41.7 | 73.2 | 65.6 |

α ± EES y β ± EES | 36.2 | 29.1 | 60.2 | 59.2 |

Predicting Moments Approach | | | | |

No change MNP | 49.8 | 37.5 | 78.4 | 73.7 |

Sk ± EES | 34.8 | 28.1 | 73.1 | 61.3 |

Std ± EES | 47.4 | 38.0 | 78.6 | 71.3 |

Xp ± EES | 48.4 | 38.8 | 81.9 | 70.5 |

Sk, Std, Xp ± EES | 30.3 | 25.9 | 59.2 | 61.6 |

The sensitivity analysis also showed the prediction approach is less sensitive (because it uses the skew coefficient) to changes in the shape parameter than the recovery approach, as average deviations reached values of 14% and 19%, respectively. However, when estimating an average deviation in the number of null hypotheses across all estimated parameters and groups of species, both procedures of parameter estimation recorded similar figures (8%).

#### 3.6. Regressing Distributional Parameters of Oaks and Pines

The average diameter and stand density of oak and pine trees, unlike the basal area, were found to be statistically related (

Figure 3). Positive statistical relationships were found between parameters of the Weibull distribution, as well (

Figure 4). The positive slope indicates an average oak diameter increment of 0.24 cm for a unit of pine diameter shift of 1.00 cm. Oak stand density was negatively related to pine stand density (

Figure 3). Oak density is 140 trees ha

^{−1} when there are no pines in the stand, but when pine density increases to 500 trees ha

^{−1}, oak density diminishes to 110 trees ha

^{−1}. When pine density increases to 1000 trees ha

^{−1}, oak density diminishes only to 81 trees ha

^{−1}. Oak density diminishes, on average, only by 6% when pine density increases 100%. That is, even though there is a statistical relationship, the slope is so small that it can be attributed to other causes of oak and pine distribution, such as subtle changes in altitude above sea level, which may modify the ratio of pine/oak diversity [

9].

**Figure 3.**
The statistical relationships between stand parameters of oaks and pines (r^{2} = coefficient of determination, Sx = Standard error, P = probability).

**Figure 3.**
The statistical relationships between stand parameters of oaks and pines (r^{2} = coefficient of determination, Sx = Standard error, P = probability).

**Figure 4.**
The statistical relationships between the Weibull distribution parameters of oaks and pines (r^{2} = coefficient of determination, Sx = Standard error, P = probability).

**Figure 4.**
The statistical relationships between the Weibull distribution parameters of oaks and pines (r^{2} = coefficient of determination, Sx = Standard error, P = probability).

#### 3.7. The Stand-Class Growth and Yield Model

The diameter distributions projected by the growth and yield model is depicted in

Figure 5. The statistical regression equations used to predict the diameter distributions are presented below in

Table 6.

**Figure 5.**
Graphical representation of the stand class growth and yield model for pines and oaks of Mexico’s northern mixed temperate forests. Capital letters indicate the sequence in mean quadratic diameter.

**Figure 5.**
Graphical representation of the stand class growth and yield model for pines and oaks of Mexico’s northern mixed temperate forests. Capital letters indicate the sequence in mean quadratic diameter.

**Table 6.**
Empirical prediction equations that form the core of the stand-class growth and yield model for pines and oaks.

**Table 6.**
Empirical prediction equations that form the core of the stand-class growth and yield model for pines and oaks.
Attributes/Species | Pinus spp. | Quercus spp. |
---|

Stand Density | 1591.5dmp^{−0.3392} (r^{2} = 0.28) | 293.15Dmq^{−0.3051} (r^{2} = 0.36) |

IDR | 30.61Np^{0.4307} (r^{2} = 0.65) | 296.9exp^{0.0007Nq} (r^{2} = 0.43) |

Dq | 1.1108Dmp^{−1.3924} (r^{2} = 0.98) | 0.69Dmq^{1.1397} (r^{2} = 0.99) |

H | 1.5158Dmp^{0.7217} (r^{2} = 0.47) | 0.9054Dmq^{0.8499} (r^{2} = 0.55) |

Cc | 69.403Dmp^{−0.1429} (r^{2} = 0.87) | 2.29Dmq^{0.8626} (r^{2} = 0.56) |

The simple allometric power-law equation fitted slightly better the H–DBH relationship than the three-parameter sigmoidal equations of Chapman-Richards or Weibull for harvested pines and oaks (

Figure 6).

**Figure 6.**
Three equations fitted to the total height—diameter at breast height relationship for pines and oaks of northern temperate forests of Mexico.

**Figure 6.**
Three equations fitted to the total height—diameter at breast height relationship for pines and oaks of northern temperate forests of Mexico.

Forest productivity in standing trees classified in forest products (sawnwood, plywood and secondary forest products) and total stand timber volume is depicted in

Figure 7.

**Figure 7.**
Graphical representation of the standing timber volume classified in merchantable forest products (sawnwood, plywood and secondary forest products) for pine and oak species of northern temperate forests of Mexico. Note capital letters indicate the sequence in mean quadratic diameter.

**Figure 7.**
Graphical representation of the standing timber volume classified in merchantable forest products (sawnwood, plywood and secondary forest products) for pine and oak species of northern temperate forests of Mexico. Note capital letters indicate the sequence in mean quadratic diameter.

Nearly 90% of the total standing volume is classified as sawnwood, about 60% is classified as plywood and less than 10% as secondary forest products (timber tips with DBH ≤ 20 cm) for both pines and oaks.

The transformation of diameter by time is derived using the equation of

Figure 8 by assuming the quadratic diameter is equal to the mean arithmetic diameter. Mean diameter is now translated into the age of trees with mean diameter and the diameter distributions transition as a function of time rather than as a function of tree dimensions. Note the diameter-age relationship for oaks is missing because a reported local one by Merlin-Bermúdez and Návar-Cháidez (41) requires further revision before it is applied in conventional forest management.

**Figure 8.**
Diameter growth curve for pines (source: 41) (Note: r^{2} = coefficient of determination, Sx = Standard error, P = probability).

**Figure 8.**
Diameter growth curve for pines (source: 41) (Note: r^{2} = coefficient of determination, Sx = Standard error, P = probability).

The growth and yield model has several weaknesses that cannot be overseen. It feeds with several empirical equations that need further revision and data to improve the variance explained, e.g., the

r^{2} ≤ 0.50. The empirical equation that predicts the shape parameter of the two-parameter maximum likelihood Weibull density function presents the smallest coefficient of determination. A different approach would be to use the empirical equation developed by Návar-Cháidez [

37] that uses only the skewness coefficient to predict α. The empirical equation to predict Sk has also one of the smallest coefficients of determination for both pines and oaks and the improvement of this equation is a matter of further research. Other stand variables must better predict either α or Sk, such as the quartiles of the diameter distribution function, basal area, site index, and altitude [

22,

39]. A parameter that explains the deviance from the normal distribution could also improve future α assessments. Regardless of these shortcomings the model predicts robust tendencies and therefore it deserves further interpretation.

The average stand density of pines and oaks diminishes as a function of mean quadratic diameter according to a power-law function. As the mean quadratic diameter increases, stand density decreases and the pine and oak diameter classes transit to the right. The stand-class growth and yield model predicts oaks appear first in these forests, while pines establish later. Most natural large disturbances, such as forest wildfires, strong cold winds, and pests and diseases open large tracts of forests. In these places and may be in other stands that are under long-term natural disturbances, oaks appear first in the stand. Oaks regenerate from sprouts in the open or in the shade and have a reproductive advantage over pines as well. Oaks left in stands after harvesting may be a second explanation for the model prediction these species remain in the forest. Pines regenerate the forest not as secondary species of succession because they are shade intolerant. They regenerate well in forest openings and at the periphery of most oak crowns. These forests are in general somehow open and sun light can reach the forest floor regardless of previously established oaks and newly regenerated pines may have access to sunlight to search for a dominant position in time. Stand density hardly surpasses 1000 trees per hectare with 10 cm in diameter at breast height (800 pines and 200 oaks). These forest dimensions point at under-stocked, open stands [

7,

54], where sun light may be available for pines to successfully regenerate in forest openings left in the forest by the small oak density. Other factors, such as soil fertility and soil water content, could be more important in promoting the regeneration of pine trees of this community.

Due to the faster diameter growth in this proposed short-term scenario, pine diameter structures transit from a J-inverted to a Gaussian shape faster than oak diameter structures. In fact, oak diameter distributions did not attain a bell-shaped distribution during this experimental scenario. However, the graphical model suggests a lack of competition processes between pines and oaks. The model predicts that a facilitation-like process takes place across the life stages of these forest stands, where partitioning of resources between pines and oaks may explain their coexistence [

55,

56]. A few oaks (

n < 200 ha

^{−1}) are left in the forest after long-term harvesting operations or natural forest disturbances or establish first because of their reproductive advantages leaving large forest openings. Pines, with a few exceptions, are shade intolerant and regenerate sexually via seed dispersal and establish well in these openings. Oaks may improve microhabitat conditions, making it possible for pines to establish successfully in open spaces in between oaks, although I had seen pines growing well beneath the canopy of large isolated oaks probably because the canopy does not interfere with sunlight entering the forest floor or perhaps because pine seedlings grow at the periphery of the canopy. Lafon

et al. [

57] recorded changes in the fertilization status of soils, given by the C:N ratio, which facilitated the establishment of pines under the canopy of oaks in east Tennessee. In some stands, pioneer pine species do not appear to establish well under the shade of oaks [

58]. Oaks, on the other side establish well under the canopy of pines as secondary species of succession or in openings as pioneer species of succession [

5].

Pioneer pine species quickly outgrow oaks in height because: (i) they are, in general, shade intolerant and grow in open spaces as well as in between the canopy of the few oaks (

n < 200 trees ha

^{−1}) present in the stand; and (ii) pines grow quickly in height searching for full sunlight to reach a dominant position in the forest. Over time, pines outgrow oaks in DBH as well, and several oak trees of most species remain dominated during the life cycle across the altitude gradient in the eastern ridges of the Sierra Madre Occidental mountain range. However, in late successional stages, several oak species attain a dominant sociological position and share this place with dominant pine trees. Therefore, the differential displacement rate of pine and oak diameter distributions may be explained by their differential growth rates and symbiotic mechanisms rather than by inter-specific competition. Domínguez and Návar [

59] supported this observation by demonstrated that by reducing 50% of stand basal area by harvesting oak trees did not improve the diameter growth of the remaining pine trees, even though pine trees were approximately 50 years-old. Therefore, resource partitioning may be playing an important role in these forests. For example, oaks and pines do not appear to compete for sunlight and it is likely they do not compete for nutrients and soil water either. That is, inter-specific competition is not as strong as it is intra-specific competition in these mixed and multiaged forests [

60]. Differential timing in the usage of resources and, most likely, the exploitation of different soil compartments, could explain the potential lack of inter-specific competition. However, further research is required on the physiological or metabolic processes of both oaks and pines to better understand as well as to put into prospective the findings of this research on the mechanisms of tree coexistence in Mexico’s northern natural forests.