1. Introduction
Tree diameter at breast height (DBH) is the fundamental attribute used to characterize individual trees and stand-level forest growth and development. It has always been used in forest management planning [
1,
2] to predict other tree- and stand-level attributes, such as tree height, tree forms, stem density, basal area, stand structure and composition, and wood biomass and volume [
3]. Model-based prediction systems for assessing forest resources require DBH at the foremost, as it is an input variable used in growth model simulators, which are used in management planning and implementation. The Forest Vegetation Simulator (FVS) [
4], Mixedwood Growth Model [
5], NATURA [
6], and the Tree and Stand Simulator [
7] are a few examples of growth simulators used in various parts of North America that rely on DBH. These simulators are used to characterize forest stand growth and predict long-term timber volumes, stand density, and basal area [
8,
9] as needed for forest management.
Silviculture practices in commercially managed forest largely depend on stand-level basal area and stem density to regulate and direct the succession trajectory throughout the rotation cycle of forest stands. The stand density management diagram [
10,
11] and the stocking diagram [
12,
13] are examples of decision tools for regulating silvicultural systems. The proper maintenance of DBH (class) distribution, stand age, structure, composition, relative frequency (or proportion) [
14], and gap dynamics [
15] are the basis for regulating forests over rotation cycles.
The DBH distribution of naturally grown forests can serve as a reference for management. For example, it helps minimize the differences of age classes in frequency distributions, while applying silviculture systems and harvest schedules. Management requires accounting for gap dynamics [
15] for competition-related mortality, natural senescence, and other forms of natural disturbance like wildfire, insect epidemics, and disease [
16,
17], in order to mimic the natural process of stand development. An appropriate silvicultural strategy can convert evenly aged forest stands into close-to-naturally grown unevenly aged structures [
18] using stem density manipulation over time. Growth models provide a useful decision support tool in long-term forest management, so more precise predictive systems are beneficial [
19].
An array of individual tree DBHs may follow some (standard) probability distribution functions (PDFs) depending on various environmental variables, such as the following dendrometric variables: stem density, basal area, dominant height, and species composition, including age structure. Geo- and topographic variables like elevation, slope, aspect, geographic position, and climate, particularly precipitation and temperature, among others, also affect species’ DBH distribution [
20,
21]. Some of these variables are static (e.g., geographic position) but many are dynamic (e.g., vegetation structure and bioclimate). Therefore, the parameter of the PDFs, once fitted, may change over time, which may be predicted by developing some predictive models [
22,
23,
24]. Alternatively, the parameters of the probability density function can be mathematically calculated by solving the system of equations of moments and percentiles [
22,
25,
26], which can be regressed in terms of dynamic variables.
Several tree species in the province of Ontario, Canada are commercially important. Although black spruce (
Picea mariana (Mill) B.S.P) and jack pine (
Pinus banksiana Lamb) are dominating species in terms of presence and abundancy across the province, 82 species are listed in provincial inventory data, of which 42 occur in our current study site in various proportions [
27]. Among these, a few have a substantial presence, depending on location and ecological distribution. These species have two sources of origin: naturally grown, mainly by wildfire disturbance, which encompasses about 75% of the sample plots or plantation origin. Naturally originated stands are richer in diversity in DBH, height, and age than the plantation origin stands [
28,
29].
There have been a few studies on the development of diameter distribution models for jack pine and black spruce grown in Ontario [
22,
30,
31,
32]. However, these models are not available for many other commercially important tree species which are less common or sub-dominating in terms of abundancy in Ontario. The study scoped that the developed model would be used in need-based forest management planning, along with being valued for use in the scientific literature for DBH simulations using simple statistics-based parameter recovery techniques. The goal of this study was to develop dynamic models for DBH distributions of eight commercial tree species grown in natural-origin mixed stands across Ontario, namely balsam fir (
Abies balsamea (L.) Mill.), eastern white pine (
Pinus strobus L.), paper birch (
Betula papyrifera Marshall), red maple (
Acer rubrum L.), red pine (
Pinus resinosa Aiton), sugar maple (
Acer saccharum Marshall), trembling aspen (
Populus tremuloides Michx), and white spruce (
Picea glauca [Moench] Voss). The specific objectives were to: (a) fit selected three PDFs, gamma, log-normal, and Weibull, for the DBHs of these species, (b) compare DBH distributions reconstructed based on the parameters obtained using maximum likelihood and parameter recovery methods, and (c) construct predictive models for the estimated parameters as functions of stand-level dynamic variables, using data collected from permanent sample plots across Ontario.
4. Discussion
DBH is a uniquely important attribute in forest research and management planning. It can be measured or estimated at the individual tree level or at a stand level. For management and harvest schedule planning, stand-level attributes are needed. Although the measurement of individual tree DBHs is common, stand-level estimates may often be more precise than individual tree level measurements, due to higher chances of error accumulation [
57]. On the other hand, stand-level DBH distribution can be used to reconstruct individual tree DBHs [
30] or size class distributions [
58].
Fitting DBH probability distributions and undertaking the reconstruction of distributions are not new techniques, but they are rarely used in practice and only for dominating species in localized geography [
24,
26]. Nonetheless, these still lack models for multiple species growing in mixed forests. We evaluated three commonly tested PDFs for eight commercial tree species in Ontario grown in natural mixed stands. The use of Weibull PDF for DBH distribution fitting is very common in a wide range of the literature [
59], with a few studies reflecting comparative analyses among competing PDFs for different species globally, e.g., [
60,
61].
We compared three PDFs: Weibull, gamma, and log-normal. Log-normal is the simplest form of distribution, after negative exponential, to fit positive continuous random variables like DBH. Our preliminary analysis indicated that the one-parametric negative exponential did not fit at all for any species. But all three PDFs generally fit well for all species except red pine. Among these PDFs, log-normal fitted the best at the landscape and sample plot levels (
Figure 2,
Table 5) for all species. In a similar study, Rijal and Sharma [
30] found that gamma PDF was the best among these three PDFs for two dominant species, jack pine and black spruce, which are also grown in Ontario. This difference could be because of the number of individual trees in the sample plots and the range of DBH. The less dominating eight species consisted of a lower number of trees within the sample plots, but the range of DBH was wider. This resulted in higher standard deviations and, hence, a longer right tail (
Figure 3 and
Figure 4) for these species, compared to the ones for jack pine and black spruce. Highly right-skewed data favor log-normal distribution, which is consistent with the previous works, e.g., [
60,
61].
We confined this study to the naturally grown forest stands. The DBH distribution of species in naturally grown forests features opposite J-shaped curves [
62]. This characteristic reverse J-shape is approximately met based on the graphical presentation for all species, with a very high negative slope at the beginning of the curves (
Figure 3). The DBH class distributions show a similar pattern for all species, except for trembling aspen and white spruce in the DBH class (0–5] and the DBH class up to (25–30] for red pine (
Figure 4). A reason for such a lower density in the DBH class below five cm might have been attributed to the undercounting of seedlings and saplings. To our knowledge, there were no recent disturbances in the area. Nevertheless, the selected PDF could fit well for trembling aspen and white spruce. In the case of red pine, it seemed that there could have been more intensive forest management intervention in the recent decades that resulted roughly uniform distribution or may require multimodal PDF. The selected eight species were sub-dominant species after jack pine and black spruce in the study area. There could be a small number of individual trees that could escape from suppression, resulting in a DBH with a long right tail, which supports log-normal distribution.
The first part of this study examined two approaches to parameter estimations, namely maximum likelihood and parameter recovery methods for the selected three PDFs, but both require solving non-linear functions. The parameter recovery technique involves stand-level attributes [
37], belonging to simple statistics, such as the mean, the standard deviation, a percentile of an array of the data, or the dominant diameter [
22,
26], among others. Among them, we used the mean and standard deviation, which are the simplest statistics and most common variables in forest management planning [
3]. Using these variables in determining distribution functions worked well compared to the maximum likelihood method (
Table 3 and
Table 5,
Figure 4). Mathematically, the maximum likelihood or moment-based methods (Equations (1)–(9)) are the same, and we should have the same results, provided there are no computational problems. Our results with the use of the selected three PDFs have demonstrated that DBH distribution can be simulated using any parametric PDFs for any range of DBHs for any species and can evaluate their performance from such simple statistics (mean and standard deviation).
From the perspective of forest management planning, our study would help the development of DBH distributions in response to various silvicultural treatments with the use of dynamic variables. The statistics of DBH change over time in response to various natural and anthropogenic disturbance factors. Consequently, either only the parameters of the PDF or the form of distribution may vary though time. More importantly, when there is intensive forest management, the growth trajectories of different species can vary substantially in a way that can be addressed with the use of simple statistics. Fitting regression models with dendrometric, geographic, and climatic variables in the second component of our work helped to update the parameters of PDF. Among the evaluated model covariates, the dendrometric variables could explain more than the selected geographic and climatic variables for most of the examined species. The parameters associated with geographic variables (longitude and latitude), as well as the annual mean and variability (standard deviation) of bioclimatic variables, were also significant for some species like balsam fir and eastern white pine when we fit of an array of DBH for eight species for mean and standard deviation (
Table 7a-b). These results correspond to previous pieces of work that showed that geography and climate can substantially impact DBH growth [
20,
49]. When dendrometric variables have a strong relationship with the response variables, like DBH, the bioclimatic variables may come to be insignificant [
32], even though these variables are always important in forest growth and development.
Stand dendrometric attributes like basal area, stem density, QMD, dominant height, and age are common as covariates in predictive modelling, e.g., [
22,
26,
30,
31]. Based on the data availability, we limited dendrometric covariates (
Table 1), and the regression evaluations demonstrated that total and species-wise stand attributes, especially QMD, stem density, and basal area, were significant covariates for predicting the mean and standard deviation for the selected species, but showed different magnitudes and signs of parameters (
Table 5). For example, total basal area as a model covariate had a negative parameter sign whenever it was significant for all species in regressions for DBH mean and standard deviation, except for the mean of trembling aspen. Stem density had a positive relationship with the DBH mean. Likewise, the QMD was positively related to the mean and the standard deviation. A reason for the opposite behaviors of the stem density and basal area could be due to the fact that the stands were dominated by small-sized trees [
63]. Such a parametric relationship with the response variable, DBH mean, was consistent with an earlier study for two dominant species: black spruce and jack pine, in the same study area [
32].
Our regression models for the DBH mean and standard deviation for eight species using various model covariates performed differently in terms of model performance statistics. In all cases, they yielded unbiased predictions. The magnitude of cross-validated other statistics, namely MAB and R2, varied by model for each species, but the median and 90 percentile range (confidence interval) values inferred that these models’ fit had precise predictions. However, there was a wide range of variations in R2, especially for the model of standard deviation. The model for the mean had a range of R2 between 0.76 and 0.91, whereas the values for the standard deviation were between zero and 0.46. The mean values are generally related linearly to the examined dendrometric covariates, resulting in a high R2. On the other hand, the standard deviation, a form of transformation of the DBH, is not linearly related to the model covariates, resulting in low R2 values despite seeing unbiased predictions.
The simulation of DBH distribution for multiple species is challenging for sub-dominating species because of their limited presence in various proportions or absence in many sample plots. There were more than 80 individual tree species listed. Modelling for all individual trees’ DBHs is neither desirable for increased complexity, nor do we achieve better estimates for very small number of sample trees. For better consistency of model parameters, we set a minimum threshold of five individual trees, and, even with this, we often encountered problems with convergence or inconsistent parameters and required several iterations for better results. Despite the possibility that other forms of PDF may still be better than ours for some species we covered, we had to limit ourselves to only these three PDFs. Nevertheless, there can be some extended avenue of improvement, leading to covering a wide spectrum by developing multi-modal distributions to capture the characteristics of the growth responses of different types of silvicultural operations in managed forests. For example, distribution fitting for red pine, in our case, may have warranted exploring multimodal PDFs, but we limited ourselves to unimodal PDFs for some resource constraints and because we hoped to keep this method operationally simple and feasible for other practitioners. A general model can be developed for minor species, in terms of their presence, by taking them together. A comparative study of the model parameters of individual trees and combined parameters for all minor species may help us to understand the species’ ecology and growth dynamics in the given forest stands. These parameters, more specific to general levels of PDF and regression fit, would help with selecting better silviculture systems for directing and maintaining forest growth and development trajectory, as desired by forest management planners and managers.