Stand density management is the process of controlling resource

Stand Density Management Diagram for Jack Pine Stands in Eastern Canada Mahadev Sharma and S.Y. Zhang ABSTRACT A stand density management diagram was developed for jack pine (Pinus banksiana Lamb.) stands using the data obtained from 125 permanent sample plots (PSPs) established in Ontario and 232 PSPs in Quebec, Canada. The diagram was evaluated using data from 40 PSPs established in Ontario. Recently developed and efficient models have been used in constructing the diagram to estimate diameters and heights for the trees for which no diameters or heights were recorded at the time of stand inventory. Relative density indices of 0.15, 0.40, and 0.55 were used, corresponding to the line of approximate crown closure, the limit of productive zone, and the lower limit of competition-related mortality, respectively. If two stand characteristics are known, including mean total tree volume, quadratic mean diameter, trees per hectare, and average dominant height, the others can be readily obtained using the diagram. The consequences of various thinning scenarios can be plotted and visualized in the field without the need for computer simulation. Keywords: mean volume, quadratic mean diameter, stand density, height and diameter isolines, maximum size density relationship Stand density management is the process of controlling resource competition through density regulation to realize specified management objectives. Stand density management diagrams (SDMDs) are innovative decision-support tools for the management of even-aged pure species stands. SDMDs are average stand-level models that graphically illustrate the dynamic relationships among stand density, tree size, and wood volume at various stages of stand development. These models can help make various forest management decisions (e.g., initial spacing, thinning, harvesting age, and regeneration) based on stand yield (wood volume). It provides resource managers with an objective method of determining density control schedules by management objectives (Newton 1997). SDMDs were developed initially by Japanese scientists in the early 1960s and subsequent modifications have been made by various researchers to apply them to different species managed with specific objectives. Newton (1997) presents a review of the historical development and applications of SDMDs for stand-level management planning. According to this review, a number of SDMDs have been developed for several commercial tree species grown in Canada. Drew and Flewelling (1977) introduced SDMDs in North American forestry literature by reviewing the self-thinning rule presented by Yoda et al. (1963). In another study, Drew and Flewelling (1979) expanded this research by calibrating an SDMD for coastal Douglas-fir (Pseudotsuga menziesii [Mirb.] Franco) plantations situated in the Pacific Northwest region of North America. Flewelling et al. (1980) developed an SDMD for western hemlock (Tsuga heterophylla [Raf.] Sarg.) stands located in the Pacific Northwest region of North America. Flewelling and Drew (1985) calibrated an SDMD for managed interior lodgepole pine (Pinus contorta var. latifolia Engelm.) stands by combining data sets from multiple regions (Oregon; BC and Alberta, Canada; Finland; Sweden; United Kingdom; and New Zealand). Similarly, Farnden (1996) produced SDMDs for lodgepole pine, white spruce (Picea glauca [Moench] Voss) and interior Douglas-fir (P. menziesii var. glauca [Beissn.] Franco.) stands grown in western Canada. Newton and Weetman (1993, 1994) developed SDMDs for both natural and managed black spruce (Picea mariana [Mill.]) stands applicable to the central and eastern Canada. Archibald and Bowling (1995) developed an SDMD for jack pine (Pinus banksiana [Lamb.]) stands using data collected from northern Ontario. Douglas-fir, black spruce, and jack pine are among the most important commercial tree species grown in Canada. As described earlier, SDMDs for Douglas-fir and black spruce were developed using the data collected from a wide range of geographic regions of Canada where these species are native. Therefore, these SDMDs can be applied to Douglas-fir and black spruce stands across the country. The SDMD for jack pine (Archibald and Bowling 1995), however, was developed using only data from northern Ontario. Consequently, prudent use of the jack pine SDMD is limited to the stands from this region. Moreover, the model components used in these SDMDs have been outdated. Recently updated and more efficient growth models are available that can be applied to develop more accurate SDMDs (Sharma and Zhang 2004b). The objective of this study was to develop an SDMD that can be applied to jack pine stands grown in eastern Canada by using more Received January 15, 2005; accepted August 5, 2005. S.Y. Zhang (Tony.Zhang@qc.forintek.ca), Resource Assessment and Utilization Group, Forintek Canada Corporation, Sainte-Foy, Quebec, Canada G1P 4R4. Mahadev Sharma (mahadev.sharma@mnr.gov.on.ca), Ontario Forest Research Institute, Ministry of Natural Resources, Sault Ste Marie, Ontario, Canada P6A 2E5. This study was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), the Living Legacy Trust, Tembec s Forestry Research Partnership, and Forintek Canada Corporation. The authors thank Mr. Dave Wood, Project Coordinator, Ontario Forest Ecosystem Science Co-operative, Inc., and Ms. Karen Zhou, Database Specialist, Ontario Ministry of Natural Resources, Quebec Ministry of Natural Resources, for providing the data used in this study. The authors also thank Mr. George Bruemmer and Mr. Al Stinson of Tembec, Mr. Murray Woods, Growth and Yield Specialist, Ontario Ministry of Natural Resources, and Mr. Yvon Grenier, Research Scientist, University of Quebec in Abitibi-Téminscamingue for their assistance throughout the study. Copyright 2007 by the Society of American Foresters. 22 NORTH. J.APPL. FOR. 24(1) 2007

efficient model components to construct the SDMD and by using data from a wide range of locations within Quebec and Ontario. Data The data used to develop the SDMD in this study (model data set) were obtained from the Ministries of Natural Resources of Quebec and Ontario, Canada. The data from the Ministry of Natural Resources of Quebec were collected from permanent sample plots (PSPs) established between 1970 and 1989 across the province. Most of the plots were established at random with plot size of 400 m 2. The plots were visited on average every 10 years resulting in two to three measurements per plot. Within each plot, the species and outside bark dbh measured 1.3 m above the ground of every tree with a dbh larger than 9 cm were recorded while the height and age of trees were measured on 1 13 randomly selected individual trees. Trees for height measurement were sampled from all diameter classes. Trees with broken top, disease, and other defects were not included in the sample. Saplings taller than 1.3 m but with a dbh smaller than 9 cm were counted by dbh classes of 2 cm within a subplot of 40 m 2 (Raulier et al. 2003). All the plots across the province of Quebec were examined for pure jack pine stands. Stands comprising a jack pine basal area (BA) of 75% or more of the total plot BA were regarded as pure jack pine stands. For the remainder of this article, a stand is regarded as a pure jack pine stand if the BA of jack pine trees is greater than or equal to 5% of total plot BA unless otherwise specified. Altogether, 232 plots qualified as pure jack pine stands from across Quebec. Among these 232 plots, 65, 108, and 59 were measured once, twice, and three times, respectively, yielding a total of 458 measurements. Similarly, the data obtained from the Ontario Ministry of Natural Resources were measured during 1993 2000 from PSPs situated in northeast and northwest regions of the province. These plots were established by the Ministry between 1992 and 1996 under a long-term comprehensive Ontario Forest Growth and Yield PSP program. The objectives of this program were to acquire and disseminate timely information and knowledge based on quantitative data related to growth, productivity, and dynamics of the forests in Ontario. The total number of plots established was 559, with plot size of 400 m 2. These plots were visited every 5 10 years, resulting in one to two measurements per plot. Within each plot, the species, origin (natural, planted, or seeded), status (live or dead), dbh (outside bark), and dbh height[1] of every tree with total height greater than 1.3 m were recorded. Crown class was recorded for every live tree. Total height and height to the crown were recorded for the trees sampled for height measurement from each plot. Some measurements in this data set were from very young stands. Therefore, stands with mean total tree volumes of less than 0.01 m 3 were not included in this study. In total, 125 plots qualified as pure jack pine stands with mean total tree volumes of 0.01 m 3 or more. Sixteen plots were measured twice and the rest were visited only once, resulting in a total of 141 measurements. The data set used for the evaluation of this SDMD (validation data set) was obtained from the Forest Ecosystem Science Cooperative, Inc., in Ontario, Canada. These data were collected by Sumac Forest Information Services from permanent growth study plots established by various forest landowners (Bowater, Weyerhaeuser, Kimberly-Clark, Domtar, Tembec, and others including the Ontario Ministry of Natural Resources). The majority of the plots are located across much of northern Ontario, Canada. The same tree and stand characteristics information was obtained for this data set using the same method as was applied to the model development data set. The total number of plots that qualified as pure jack pine stands in this case was 40. Methods As outlined by Newton and Weetman (1993), the approach used in the development of an SDMD for jack pine consisted of the following procedures: 1. Determination of the mean volume (MV) density relationship and relative density index (Drew and Flewelling 1979) corresponding to i. The asymptotic MV density condition ( 3/2 power law of self-thinning; Yoda et al. [1963]). ii. The lower limit of the imminent competition mortality zone (Drew and Flewelling 1979). iii. The productive zone limit. iv. The approximate crown closure. 2. Derivation of dominant height (DHT) and quadratic mean diameter isolines. For each plot measurement, the following computations were made to estimate the tree and stand characteristics required for this SDMD. Tree Heights Diameter height relationship depends on stand density (Sharma and Zhang 2004b). Sharma et al. (2002) reported that both height and dbh growth are affected by density but the effect is less on height than on dbh. For a given dbh, therefore, a tree generally is taller when grown at a higher density than at a lower density (Sharma and Zhang 2004a). Consequently, the height of a tree expressed in terms of diameter without incorporating the density effect may not be accurately estimated. Sharma and Zhang (2004b) developed height diameter relationship models by incorporating stand characteristics such as stand BA per hectare and trees per hectare (TPH) to estimate the heights of jack pine and black spruce trees more accurately. These models were developed using the same data sets used in this study. Sharma and Zhang (2004b) found these models superior to the previously developed models in estimating the total height of these tree species. Therefore, total heights of the trees were calculated using these models in terms of stand BA per hectare, TPH, and dbh for the trees not measured for total height. For most of the pure jack pine stands used in this study, black spruce contained the second largest BA. The combined BAs of jack pine and black spruce accounted for up to 95% of the total stand BA. Therefore, separate equations were used to estimate the heights of jack pine and black spruce trees. These equations developed by Sharma and Zhang (2004b) were for jack pine H 1.3 5.2067(BA) 0.4177 1 e 2.428(TPH) 0.3292 (dbh) ) (1) for black spruce H 1.3 14.6726(BA) 0.2176 1 e 0.1175 TPH) 0.1144 (dbh (2) NORTH. J.APPL. FOR. 24(1) 2007 23

Table 1. Parameter estimates (standard errors in parentheses) for Equation 3 fit to jack pine and black spruce diameter-age data using nonlinear regression. Parameter Jack pine Black spruce M 104.1 (23.1293) 70.7012 (5.7937) 0.5114 (0.0244) 0.7537 (0.0196) MSE 0.6486 0.8069 R 2 * 0.98 0.96 *Computed as (1 residual sum of squares/corrected sum of squares). where H is total height of a tree (m), BA is stand BA (m 2 /ha), TPH is stand density (trees/ha), and dbh is diameter at breast height (cm). Because the presence of other tree species (balsam fir (Abies balsamea [L.] [Mill.]), white birch (Betula papyrifera Marsh.), white spruce (Picea glauca [Moench] Voss.), and trembling aspen (Populus tremuloides Michx.) in the pure jack pine stands was not significant, heights of trees of other species were estimated using the equation developed for jack pine trees. Outside Bark dbh The Quebec data only contained the diameter measurements for trees greater than 9 cm. Therefore, if a tree had a dbh smaller than 9 cm at the first inventory and was 9 cm or more at the second inventory, the tree was not measured for dbh during the first inventory even though it was present in the stand. If, in developing an SDMD, we do not include the trees that were not qualified for measurement at a certain inventory but were measured at a subsequent inventory, the result from the SDMD would not be consistent. As a result, the dbh of the trees that were not measured at certain inventories needed to be estimated. A diameter growth model was used to estimate the dbh of a tree at a certain inventory given the diameter and age at another inventory. The following model form (McDill and Amateis 1992) was used for this purpose: D 1 M 1 1 M D 2 A 2 A 1 (3) where D 1 is the dbh at age A 1, D 2 is the dbh at age A 2, M is the maximum value of the diameter to be achieved (asymptote), and is the growth parameter. The parameters of this model were estimated for jack pine and black spruce trees by fitting the model to the diameter-age data of the model data set of these tree species separately. Parameter estimates and their fit statistics are presented in Table 1. Total Tree Volume, BA, and DHT Total tree volumes of all live trees for all species were calculated using the respective total tree volume equations developed by Honer et al. (1983). BA also was calculated for each live tree. These volumes and BAs were summed to compute the total volume and BA, respectively, for each stand. The quadratic mean diameter for each stand was calculated from total BA and the number of live trees in the stand. DHT was computed as the average height of all dominant and codominant trees recorded at the time of measurement. Estimated heights of dominant and codominant trees were not included in computing average DHT. Table 2 presents the summary statistics of the stand characteristics used in this study. To compute stand characteristics, average values for TPH, BA per hectare, volume per hectare, quadratic mean dbh (QMDBH), Table 2. the study. Summary statistics of the stand characteristics used in Variable Mean SD Minimum Maximum Ontario permanent sample plot data (n 141) TPH 1329 673.03 275 4225 BA/ha (m 2 /ha) 26.28 7.78 7.61 42.56 Volume/ha (m 3 /ha) 87.38 44.22 10.53 252.71 DHT (m) 18.04 3.88 8.98 24.38 QMDBH* (cm) 17.00 4.14 8.99 25.65 Age (yr) 62.48 23.39 10 127 Quebec permanent sample plot data (n 458) TPH 1298 617.41 200 2950 BA/ha (m 2 /ha) 17.91 9.49 1.13 44.78 Volume/ha (m 3 /ha) 121.10 89.40 3.16 502.84 DHT (m) 13.31 3.49 5.82 25.90 QMDBH* (cm) 13.28 3.02 8.48 26.29 Age (yr) 54.26 20.51 17.00 151.00 Evaluation data set from Ontario (n 40) TPH 1414 830 400 3325 BA/ha (m 2 /ha) 20.32 6.55 5.90 31.88 Volume/ha (m 3 /ha) 141.89 58.06 19.25 254.37 DHT (m) 14.84 3.72 6.12 21.42 QMDBH* (cm) 14.70 4.03 6.97 24.69 Age (yr) 66.93 16.93 39.00 94.00 *QMDBH (outside bark). and average DHT were first calculated based on individual tree values within a plot. These plot averages were then used to calculate the overall average for each stand characteristic for each data set (Table 2). Overall average values for TPH, BA per hectare, DHT, QM- DBH, and stand age were slightly higher for the Ontario data set than for the Quebec data set. Volume per hectare, however, was higher for the Quebec data set than for the Ontario data set. Also, ranges for all stand characteristics except TPH were greater for the Quebec data set than for the Ontario data set. Relative Density Index To check if the data from these stands could be used to estimate a maximum size density relationship for jack pine trees, mean total tree volume was plotted against TPH on a log scale (Figure 1). It is clear that a considerable number of stands were in the development stage (no clear evidence of competition-related mortality). The rest of the stands were fully developed and hence appeared to be showing competition-related mortality. To confirm this assertion, another plot was made for mean total tree volume against TPH for the stands that were measured at least twice (Figure 2). As this figure shows, the volume density trajectories for the mature stands were found to follow the hypothetical line approximately representing the maximum size density relationship. The maximum size density relationship line ( 3/2 power law of self-thinning) was estimated by computing a loss function for the line using SAS (SAS Institute 1999). The loss function was calculated by applying the Marquardt method to logarithmically transformed average total tree volumes and densities using nonlinear regression analysis techniques. The maximum size density relationship line was referred to the line corresponding to relative density index, P r 1.00 (Drew and Flewelling 1979). Other lines corresponding to P r 0.55, 0.40, and 0.15 were chosen as the lower limit 24 NORTH. J.APPL. FOR. 24(1) 2007

Figure 1. Scatterplot of mean total tree volume (m 3 ) against TPH from all stands in Ontario and Quebec. of competition-related mortality, the limit of productive zone, and the line of approximate crown closure, respectively, as explained by Drew and Flewelling (1979) and Grenier and Harvey (2004). Diameter and DHT Isolines The relationships of QMDBH and average DHT isolines to MV and stand density (TPH) were assigned using the following equations: log 10 (MV) log 10 (QMDBH) log 10 (TPH) (4) 1 MV DHT (TPH) DHT (5) where,,, and are parameters. Parameters for Equation 4 were estimated using linear regression analysis procedures on logarithmically transformed data. Parameter estimates for Equation 5, however, were obtained using nonlinear regression techniques in SAS. These estimates along with their fit statistics are presented in Table 3. All these components were assembled in a spreadsheet to construct an SDMD for jack pine. The self-thinning line, the lower limit of the zone of imminent competition-related mortality, the limit of productive zone, the approximate crown closure line, QMDBH, and average DHT and codominant height isolines were superimposed on a bivariate graph with MV on the ordinate axis and stand density on the abscissa to develop an SDMD for jack pine trees (Figure 3). NORTH. J.APPL. FOR. 24(1) 2007 25

Figure 2. Mean total tree volume plotted against the TPH for the stands that were measured at least twice. Table 3. Parameter estimates (standard errors in parentheses) fit to logarithmically transferred data using linear regression for Equation 4 and fit to untransformed data using nonlinear regression for Equation 5. Parameter Equation 4 Equation 5 5.0905 (0.0205) 1638.0 (823.4) 3.0696 (0.0121) 2.0606 (0.2154) 0.1859 (0.0046) 0.5487 (0.5465) 1.8877 (0.4637) MSE 0.00021 32.1431 R 2 * 0.9973 0.7149 *Computed as (1 residual sum of squares/corrected sum of squares). The jack pine SDMD was evaluated using the validation data set (independent from the model data set) obtained from Ontario. Predicted values of mean total tree volume were computed using Equations 4 and 5 and bias (observed predicted) was calculated for each equation. Evaluation was performed by examining the bias in estimating the mean total tree volume. Results and Discussion Like height diameter models, the diameter growth models obtained by fitting Equation 3 to jack pine and black spruce tree data were very efficient (R 2 0.98 and 0.96, for jack pine and black 26 NORTH. J.APPL. FOR. 24(1) 2007

Figure 3. Jack pine stand density management diagram graphically illustrating (1) self-thinning line at a relative density index P r 1.0, lower limit of the zone of imminent competition mortality at P r 0.55, limit of productive zone at P r 0.40, approximate crown closure line at P r 0.15, and QMDBH and average DHT and codominant height isolines. spruce, respectively). The estimates for the rate and asymptote parameters for jack pine were significantly different from their counterparts for black spruce (Table 1). Therefore, separate models were used to estimate the diameters of these tree species. Because the contribution of the other tree species to the stand BA was negligible, models fitted to the jack pine data were used to estimate the diameters of the rest of the tree species present in the stands. Drew and Flewelling (1979) and Weller (1987) have already pointed out that there is no rigorous statistical procedure available for selecting the limiting boundary of a zone, given that some unknown random variation is to be expected. Therefore, Drew and Flewelling (1979) chose the maximum size density relationship line near the upper limit of the data with the slope of 3/2. Other researchers, however, have used regression as well as principal component analyses to fit the line through the data near the boundary with standard least squares techniques (Weller 1987). The estimates for the slope and intercept of the maximum sizedensity relationship line were 1.45 and 4.049, respectively, in the logarithmic (base 10) scale, i.e., the maximum size density relationship line was log 10 (volume) 4.049 1.45 log 10 (TPH) (6) This line was referred to as the line corresponding to a relative NORTH. J.APPL. FOR. 24(1) 2007 27

Figure 4. Jack pine stand density management diagram with DHT isolines calculated from the model, MV DHT (TPH) (DHT). density index, P r 1.0. According to Drew and Flewelling (1979) relative density indices can be interpreted as follows: A stand with a relative density index less than 0.15 will have growth per unit area proportional to the density. At relative densities between 0.15 and 0.40, growth per unit area increases with density, but growth per tree declines. At relative densities between 0.40 and 0.55, growth per unit area is unaffected by density. For stands with relative densities greater than 0.55, gross growth is the same as in the 0.40 0.55 region, but net growth may be considerably less if substantial mortality has occurred. Table 3 displays the parameter estimates and fit statistics for Equations 4 and 5 using linear and nonlinear regressions, respectively. The fit for QMDBH against mean total tree volume (MV) and TPH was almost perfect (R 2 0.997). Therefore, QMDBH was expressed in terms of MV and TPH using the relationship 28 NORTH. J.APPL. FOR. 24(1) 2007

defined by Equation 2 to draw diameter isolines on the diagram. Relationships among DHT, TPH, and MV through Equation 5, however, resulted in a coefficient of determination (R 2 ) of 0.71. Therefore, other forms of relationships among these variables were investigated also in a search for a better model. Dependent and independent variables were transformed by applying several functions, e.g., log, exponential, inverse, and others. These transformed and untransformed variables were combined in all possible biologically meaningful manners and fitted to the data set. The model with dependent variable MV rather than 1/MV in Equation 5 resulted in a better fit with a coefficient of variation (R 2 ) of 0.89. The DHT isolines using this model, however, were inconsistent for densities greater than 1,100 TPH as shown in Figure 4. Therefore, Equation 5 was selected to produce DHT isolines on the diagram. The DHT isolines determined using this equation resulted in consistent results as shown in the diagram. Because MV was expressed in terms of QMDBH and TPH in Equation 4 and as a function of DHT and TPH in Equation 5, bias in estimating MV was calculated using both of these equations for each plot. The mean value of the bias was 0.001 for Equation 4 and 0.028 for Equation 5 with SD 0.009 and 0.040, respectively. This was expected because the fit statistics for Equation 4 were much better than their counterparts for Equation 5. Therefore, if QM- DBH is available, MV should be estimated using Equation 4. Conclusions An SDMD was developed for jack pine trees grown in Ontario and Quebec, Canada. In constructing the diagram, newly developed and more efficient models have been used to estimate diameters and heights for trees for which these parameters were not recorded at the time of stand inventory. Relative density indices of 0.15, 0.40, and 0.55 were used corresponding to the line of approximate crown closure, the limit of productive zone, and the lower limit of competition-related mortality, respectively. This diagram will allow forest managers to formulate reasonable consequences to various density manipulations of even-aged jack pine stands. If two stand characteristics we know including mean total tree volume, quadratic mean diameter, TPH, and average DHT, the others can be readily obtained using the diagram. The diagram can be used to find out at what heights, diameters, and densities the stands start to close and subsequently incur mortality. The consequences of various thinning scenarios can be plotted and visualized in the field without the need for computer simulation. Endnote [1] If swellings, bumbs, depressions, or branches occur 1.3 m aboveground, tree diameters usually are measured above or below the breast height. Literature Cited ARCHIBALD, D.J., AND C. 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