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Abstract Through the study of parse wood materials in Shandong Province， the fitting empirical equation of tree growth was obtained， a function with tree growth was a variable and time as an independent variable. Through mathematical operations such as function derivation， the mature age of tree growth was obtained. The obtained expected maturity age for Populus canadensis Moench forest was 11 a for pulp material， 26 a for pillar materials and 41 a for peeler. And the application， research directions and precautions of the mature ages were proposed.
Key words Expected maturity age； Empirical equation； Parse wood
In forestry production， the formulation of cutting quotas and cutting area design must first meet the problem of the maturity age of trees. However， Shandong Forestry has done less work on the basics of the number table. Most of them use foreign or national standards， and do not change for decades， which will inevitably cause great deviations. In this paper， by using the data of parse wood materials in Shandong Province， the maturity ages of Populus canadensis Moench forest were studied. P. canadensis was the main tree species in plain afforestation in Shandong Province from the 1960s to the 1980s. However， with the advent of new black poplar varieties， it has been eliminated. Because the growth rate of black poplar species is better than other broad??leaved tree species， vigorously creating black poplar trees is of great significance to the production of wood and the adjustment of agricultural planting structure. Exploring the maturity age of P. canadensis is very useful for guiding the production of black poplar species.
Source
Due to limited funding， the previous survey materials were used in the paper. The parse wood materials were collected from a dominant 25??year??old tree of P. canadensis with normal growth from Shahe Forest Farm in Shan County on March 31， 1985. The diameter at bread height （DBH） was measured in the section of 2.6 m， and other parameters were measured in the section of 2 m. Round circles were intercepted a the tree height of 5 cm （circle 0）， 1.3 ， 3.6 ， 5.6 ， 7.6 ， 9.6， 11.6 and 12.6 m， and the circles were strictly interpreted in accordance with the technical requirements of Parsing Wood. Relevant information was collected with the age??class of 1 year.
Research methods
In order to save research costs， based on the analysis of parse wood data， the tree growth empirical equation was used to conduct fitting tests on various regression equations according to previous research methods[1-3]by referring to previous research methods and processes[4-5]and research results[6]. The empirical equation of tree growth was established， and various regression equations were fitted. . Finally， the following mixed empirical equation was adopted to study the growth of trees： y（t） =ea-b/t
Where， a， b are the exponential parameters of the function to be solved； e is the base of natural logarithm 2.718 28...）.
The growth of trees is affected by various factors， but the factor with the greatest impact on P. canadensis is the precipitation volume and uniformity of spatial and temporal distribution. Based on the empirical equations to fit the process of tree growth， the numerical mature age of ground diameter growth was obtained by getting the maximum age from ground diameter fitting equation （including the equations generated by the derivatives， expressed in the research process）， the mature age of tree height growth was obtained by getting the maximum age from the tree height fitting equation. The numerical mature ages for the growth of DBH， DBH square， tree height， wood volume were obtained in the same way.
Research process
A linear equation was obtained by taking the logarithm of the tree growth equations， which was then used to get the values of parameters a， b. The F??test and correlation coefficient R test of the 2 parameters were performed[2]. Through the tests， the tree growth fitting equations was established （Table 1）. As shown in Table 1， in addition to the fitting equations of tree height and wood value 1， which passed the F??test with the reliability of 90%， the fitting equations for other items all passed the F??test and R??test with the reliability over 99%， indicating that this mathematical model （the empirical fitting equation） was applicable as a whole. All items passed the correlation coefficient R test with reliability of 99.9%， suggesting that the fitting equation relationship was established. The maximum time of current annual increment and numerical maturity age of trees by the fitting equations were illustrated with ground diameter 1 as an example. For the equation of average growth rate of trees （current annual increment was completed by the derivation of the function Y（t） in Table 1， and only the extreme point was given in the paper）， the extreme point tz=4.25 a， that is， the current annual increment reached the peak when the tree reached 4 years old， and the peak was a single one. For the equation of numerical maturity age of trees （annual average increment， and only the extreme point was given in the paper）， the extreme point tm=8.49 a， so the numerical maturity age of the trees was 8.49 a. In this paper， only the fitting equations for ground diameter were stated， and all other fitting equations were done in the same way. The meanings were all the same for growth fitting equation， tree growth rate equation， tree average growth speed equation， so were the significances of symbols of tz， tm， so the calculation results were given indirectly in the paper. The numerical maturity ages of each item were shown in Table 1. As shown in Table 1， the squares of the indices were doubled compared with the indices， but the accuracies were equal to the F??test values and R??test values. This was caused by the exponential mathematical relationship. In order to compare with the accumulation fitting equation， a fitting equation of DBH square was deliberately established. In Table 1， the values of tn were the ages at the points of intersections of the curves of current annual increments and annual average increments of the sample wood （the values of wood volume 2 were obtained using the tendency chart of growth curves）， which could be used as the actual maturity ages of the tree wood. The tm values of ground dimater， DBH 1， DBH 2， tree height and wood volume were close to the values of tn， while the tm and tn values of DBH square showed significant proximities to the values of wood volume 2， and so this study was of representative significance. Therefore， it was more suitable to fit the wood volumes using empirical equations. The research results had high reliability. In this paper， the maturity of trees was considered according to the maturity of tree volume. Thus， based on the research results and the needs for production practice， it was more suitable to set the maturity age of accumulation volume as the maturity age of the tree. The fitting results of wood volumes from the calculation of empirical equations were consistent with the fitting results obtained from the parse wood analysis， thereby indicating that the accuracy and applicability of the test were reliable[3]. Conclusion and Application
Different from previous studies on P. tomentosa， 2 different maturity ages were obtained in this study （3 maturity ages for P. tomentosa）， namely 26 years for small diameter timer and 36 years for middle and large diameter timer. The difference was probably caused by the fact that the sample tree in this study was the average tree， while the sample tree for the study on P. tomentosa was dominant tree species. The differences in the growth vigor resulted in the different results. The division of the age groups was shown in Table 2.
The original standard was 26 years. Compared with the results of this study， the maturity age for small diameter timer was 21 years， which was advanced by 5 years from the previous average tree fitting equation and original standard， accounting for 19.2%. Moreover， the maturity age for large diameter timer was also advanced by 5 years from the results of average tree， accounting for 13.9%. Therefore， the acceleration of tree growth speed promoted the maturity of trees， which was consistent with the general rules for tree growth and development.
Discussion
The original maturity standard was 26 a， which was based on the standard set at the age??class of 5 years. In this study， the age??class was 1 year old. In order to take care of the production convenience， the maturity age is still set at 26 a， and therefore it is closer to reality. In this study， the concept of 2 maturity ages is proposed， so it can determine the afforestation density， business activities， and management intensity reasonably according to different business purposes. Therefore， with the passage of time， it should make appropriate adjustments with the times according to the actual situation in the local area. This study successfully solves the problem of flexible use of dimension and points out the applicability of the dimension change， especially the reduction of wood volume with dimension. The value of energy parameter b value and the fitting accuracy of the equation are unchanged， only the value of generation a shows regular changes. Table 1 only shows the fitting equations with high test accuracies and close to the actual. Thus， it makes it possible to get the expected maturity ages of tree wood through equation fitting according to the analysis on the trends of growth curves with the lack of the observation data of trees with the age??class of 26 years （and above）. Compared with the other average trees of the same tree species in the same site， the maturity age of this study is advanced by 5 years， and so it is of great practical guiding significance for the reasonable organization of selective cutting operations. This study has shown strong similarities and suitability of equations with the previous research on Querus acutissima， Rpbinia pseudoacacia， and conifers， and it can be applied in future production practice. Due to the limitations of various conditions， various deviations are unavoidable， which can only be improved and developed in the subsequent production and scientific research practices. The proposed forestry production proposals only represent personal opinions. After all， the suggestions are concluded from the analysis and fitting of an individual plant of parse wood material， and it needs the verification and approval of experts and the production practice to put them into application. The results of this study are only useful for the production of black poplar trees. In order to guide forestry production more reasonably， it is necessary to analyze the growth process of black poplar trees， which requires further research.
References
[1] Northeast Forestry University. Forest measurement[M]. Northeast Forestry University， 1985： 283-296.
[2] Northeast Forestry University. Mathematical statistics[M]. Beijing： China Forestry Publishing House， 1985： 205-251.
[3] LIU GJ. Registered consulting engineer （investment） qualification examination materials review guidance[M]. Tianjin， Tianjin University Press， 2003： 231-246.
[4] HU HY. Research on the actual maturity of individual Pinus densiflora[J]. Journal of Shandong Forestry Science and Technology， 2010， 6： 36-37.
[5] LI LP， DONG HF， ZHANG HB， et al. Study on anticipant mature age of Pinus densiflora in Shandong Province[J]. Journal of Anhui Agricultural Science， 2017， 3： 184-186.
[6] GAO JH. Approach into desirable period of forest management in Shandong Province[J]. China Forestry Science and Technology， 2003， 3：6-8.
Key words Expected maturity age； Empirical equation； Parse wood
In forestry production， the formulation of cutting quotas and cutting area design must first meet the problem of the maturity age of trees. However， Shandong Forestry has done less work on the basics of the number table. Most of them use foreign or national standards， and do not change for decades， which will inevitably cause great deviations. In this paper， by using the data of parse wood materials in Shandong Province， the maturity ages of Populus canadensis Moench forest were studied. P. canadensis was the main tree species in plain afforestation in Shandong Province from the 1960s to the 1980s. However， with the advent of new black poplar varieties， it has been eliminated. Because the growth rate of black poplar species is better than other broad??leaved tree species， vigorously creating black poplar trees is of great significance to the production of wood and the adjustment of agricultural planting structure. Exploring the maturity age of P. canadensis is very useful for guiding the production of black poplar species.
Source
Due to limited funding， the previous survey materials were used in the paper. The parse wood materials were collected from a dominant 25??year??old tree of P. canadensis with normal growth from Shahe Forest Farm in Shan County on March 31， 1985. The diameter at bread height （DBH） was measured in the section of 2.6 m， and other parameters were measured in the section of 2 m. Round circles were intercepted a the tree height of 5 cm （circle 0）， 1.3 ， 3.6 ， 5.6 ， 7.6 ， 9.6， 11.6 and 12.6 m， and the circles were strictly interpreted in accordance with the technical requirements of Parsing Wood. Relevant information was collected with the age??class of 1 year.
Research methods
In order to save research costs， based on the analysis of parse wood data， the tree growth empirical equation was used to conduct fitting tests on various regression equations according to previous research methods[1-3]by referring to previous research methods and processes[4-5]and research results[6]. The empirical equation of tree growth was established， and various regression equations were fitted. . Finally， the following mixed empirical equation was adopted to study the growth of trees： y（t） =ea-b/t
Where， a， b are the exponential parameters of the function to be solved； e is the base of natural logarithm 2.718 28...）.
The growth of trees is affected by various factors， but the factor with the greatest impact on P. canadensis is the precipitation volume and uniformity of spatial and temporal distribution. Based on the empirical equations to fit the process of tree growth， the numerical mature age of ground diameter growth was obtained by getting the maximum age from ground diameter fitting equation （including the equations generated by the derivatives， expressed in the research process）， the mature age of tree height growth was obtained by getting the maximum age from the tree height fitting equation. The numerical mature ages for the growth of DBH， DBH square， tree height， wood volume were obtained in the same way.
Research process
A linear equation was obtained by taking the logarithm of the tree growth equations， which was then used to get the values of parameters a， b. The F??test and correlation coefficient R test of the 2 parameters were performed[2]. Through the tests， the tree growth fitting equations was established （Table 1）. As shown in Table 1， in addition to the fitting equations of tree height and wood value 1， which passed the F??test with the reliability of 90%， the fitting equations for other items all passed the F??test and R??test with the reliability over 99%， indicating that this mathematical model （the empirical fitting equation） was applicable as a whole. All items passed the correlation coefficient R test with reliability of 99.9%， suggesting that the fitting equation relationship was established. The maximum time of current annual increment and numerical maturity age of trees by the fitting equations were illustrated with ground diameter 1 as an example. For the equation of average growth rate of trees （current annual increment was completed by the derivation of the function Y（t） in Table 1， and only the extreme point was given in the paper）， the extreme point tz=4.25 a， that is， the current annual increment reached the peak when the tree reached 4 years old， and the peak was a single one. For the equation of numerical maturity age of trees （annual average increment， and only the extreme point was given in the paper）， the extreme point tm=8.49 a， so the numerical maturity age of the trees was 8.49 a. In this paper， only the fitting equations for ground diameter were stated， and all other fitting equations were done in the same way. The meanings were all the same for growth fitting equation， tree growth rate equation， tree average growth speed equation， so were the significances of symbols of tz， tm， so the calculation results were given indirectly in the paper. The numerical maturity ages of each item were shown in Table 1. As shown in Table 1， the squares of the indices were doubled compared with the indices， but the accuracies were equal to the F??test values and R??test values. This was caused by the exponential mathematical relationship. In order to compare with the accumulation fitting equation， a fitting equation of DBH square was deliberately established. In Table 1， the values of tn were the ages at the points of intersections of the curves of current annual increments and annual average increments of the sample wood （the values of wood volume 2 were obtained using the tendency chart of growth curves）， which could be used as the actual maturity ages of the tree wood. The tm values of ground dimater， DBH 1， DBH 2， tree height and wood volume were close to the values of tn， while the tm and tn values of DBH square showed significant proximities to the values of wood volume 2， and so this study was of representative significance. Therefore， it was more suitable to fit the wood volumes using empirical equations. The research results had high reliability. In this paper， the maturity of trees was considered according to the maturity of tree volume. Thus， based on the research results and the needs for production practice， it was more suitable to set the maturity age of accumulation volume as the maturity age of the tree. The fitting results of wood volumes from the calculation of empirical equations were consistent with the fitting results obtained from the parse wood analysis， thereby indicating that the accuracy and applicability of the test were reliable[3]. Conclusion and Application
Different from previous studies on P. tomentosa， 2 different maturity ages were obtained in this study （3 maturity ages for P. tomentosa）， namely 26 years for small diameter timer and 36 years for middle and large diameter timer. The difference was probably caused by the fact that the sample tree in this study was the average tree， while the sample tree for the study on P. tomentosa was dominant tree species. The differences in the growth vigor resulted in the different results. The division of the age groups was shown in Table 2.
The original standard was 26 years. Compared with the results of this study， the maturity age for small diameter timer was 21 years， which was advanced by 5 years from the previous average tree fitting equation and original standard， accounting for 19.2%. Moreover， the maturity age for large diameter timer was also advanced by 5 years from the results of average tree， accounting for 13.9%. Therefore， the acceleration of tree growth speed promoted the maturity of trees， which was consistent with the general rules for tree growth and development.
Discussion
The original maturity standard was 26 a， which was based on the standard set at the age??class of 5 years. In this study， the age??class was 1 year old. In order to take care of the production convenience， the maturity age is still set at 26 a， and therefore it is closer to reality. In this study， the concept of 2 maturity ages is proposed， so it can determine the afforestation density， business activities， and management intensity reasonably according to different business purposes. Therefore， with the passage of time， it should make appropriate adjustments with the times according to the actual situation in the local area. This study successfully solves the problem of flexible use of dimension and points out the applicability of the dimension change， especially the reduction of wood volume with dimension. The value of energy parameter b value and the fitting accuracy of the equation are unchanged， only the value of generation a shows regular changes. Table 1 only shows the fitting equations with high test accuracies and close to the actual. Thus， it makes it possible to get the expected maturity ages of tree wood through equation fitting according to the analysis on the trends of growth curves with the lack of the observation data of trees with the age??class of 26 years （and above）. Compared with the other average trees of the same tree species in the same site， the maturity age of this study is advanced by 5 years， and so it is of great practical guiding significance for the reasonable organization of selective cutting operations. This study has shown strong similarities and suitability of equations with the previous research on Querus acutissima， Rpbinia pseudoacacia， and conifers， and it can be applied in future production practice. Due to the limitations of various conditions， various deviations are unavoidable， which can only be improved and developed in the subsequent production and scientific research practices. The proposed forestry production proposals only represent personal opinions. After all， the suggestions are concluded from the analysis and fitting of an individual plant of parse wood material， and it needs the verification and approval of experts and the production practice to put them into application. The results of this study are only useful for the production of black poplar trees. In order to guide forestry production more reasonably， it is necessary to analyze the growth process of black poplar trees， which requires further research.
References
[1] Northeast Forestry University. Forest measurement[M]. Northeast Forestry University， 1985： 283-296.
[2] Northeast Forestry University. Mathematical statistics[M]. Beijing： China Forestry Publishing House， 1985： 205-251.
[3] LIU GJ. Registered consulting engineer （investment） qualification examination materials review guidance[M]. Tianjin， Tianjin University Press， 2003： 231-246.
[4] HU HY. Research on the actual maturity of individual Pinus densiflora[J]. Journal of Shandong Forestry Science and Technology， 2010， 6： 36-37.
[5] LI LP， DONG HF， ZHANG HB， et al. Study on anticipant mature age of Pinus densiflora in Shandong Province[J]. Journal of Anhui Agricultural Science， 2017， 3： 184-186.
[6] GAO JH. Approach into desirable period of forest management in Shandong Province[J]. China Forestry Science and Technology， 2003， 3：6-8.