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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 tomentosa 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 tomentosa forest were studied.
P. tomentosa is the main tree species in plain afforestation in Shandong Province. Its growth rate is superior to that of other broadleaved tree species， and vigorously creating P. tomentosa forests is of farreaching and great significance for the improvement of plain afforestation. Therefore， it is necessary to explore the maturity age of P. tomentosa.
Material Sources
Due to the limited resources of P. tomentosa and limited funding， the previous survey materials were used in the paper. The parse wood materials were collected from a dominant 25yearold tree of P. tomentosa 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 ageclass of 1 year.
Research Methods
In order to save costs， the tree growth empirical equation was used to conduct fitting tests on various regression equations according to Forest Measurement[1]， a textbook edited by Northeast Forestry University in 1985. 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. tomentosa 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 Ftest 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 Ftest with the reliability of 90%， the fitting equations for DBH 1 and its square， tree height， wood volume 2 and wood volume 3 all passed the Ftest with the reliability over 95%， and the fitting equations for others passed the Ftest and Rtest with the reliability over 99%， indicating that this mathematical model （the empirical fitting equation） was applicable as a whole. All 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 = 8.05a， that is， the current annual increment reached the peak when the tree reached 8 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 = 16.11 a， so the numerical maturity age of the trees was 16.11 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 indexes were doubled compared with the indexes， but the accuracies were equal to the Ftest values and Rtest 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 3 were obtained using the tendency chart of growth curves）， which could be used as the actual maturity ages of the tree wood. In addition to DBH 1， DBH 2， which showed great differences from their squares tm and tn， all other were very close to the values of tn. 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]. Conclusions and Applications
Different from previous studies， 3 different maturity ages were obtained， namely 11 years， 26 years and 41 years. Baased on the production practice， the 3 maturity ages were named as the maturity age for pulpwood （or short rotation）， maturity age for pillar wood （beam） and maturity age for peeler. The maturity age obtained from the fitting equation of DBH 1 was very close to the fitting value of wood volume 1， which was consistent with the results obtained from the previous traditional method. The maturity age obtained from the square of DBH 2 was very close to the fitting value obtained from the wood volume 2， which proved the preciseness of the research method. The test results were very close to the actual values. The division of the age groups was shown in Table 2.
The original standard was 9 years for pulpwood and 26 years for pillars， which was basically consistent with the conclusions of this study， but the standard for peeler has not yet been reported.
Discussion
The original maturity standard was 26 a， which was based on the standard set at the ageclass of 5 years. In this study， the ageclass 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 3 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 ageclass of 26 years （and above）. From this point of view， the issues concerning the maturity ages of trees cannot be the same. If conditions allow， some fitting equation tests can be conducted to find out more suitable maturity ages for the local so as to better guide the forestry production， which is of great importance. Due to the difficulty in collecting tree samples and limited funding， the parse wood materials were used， and the fitting empirical equation of tree growth was used to make up for the lack of the ages of parse wood， which also avoids the noise effects on the test results from the spatial differences and tree differentiation under various natural conditions. The maturity ages of the actual P. tomentosa forest are obtained in this way. After repeated verification， analysis and judgment， the empirical equation is applicable for the tree growth， but it is hard to make scientific explanation. Most of the observed effective figures in this study reach triple digits， which ensures the research accuracy objectively， so the method can be put into production practice for promotion and application. The improvement of the accuracy of observed data can make up for the insufficient ageclass data， so as to save the research expenditures. As for the accuracy of observation data， it should be improved in the subsequent scientific practice to ensure that the effective number of observation data reaches more than triple digits. 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.
Changgang WANG et al. Discussion on Individual Expected Maturity Age of Populus tomentosa Carr
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.
Editor： Na LI Proofreader： Xinxiu ZHU
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 tomentosa forest were studied.
P. tomentosa is the main tree species in plain afforestation in Shandong Province. Its growth rate is superior to that of other broadleaved tree species， and vigorously creating P. tomentosa forests is of farreaching and great significance for the improvement of plain afforestation. Therefore， it is necessary to explore the maturity age of P. tomentosa.
Material Sources
Due to the limited resources of P. tomentosa and limited funding， the previous survey materials were used in the paper. The parse wood materials were collected from a dominant 25yearold tree of P. tomentosa 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 ageclass of 1 year.
Research Methods
In order to save costs， the tree growth empirical equation was used to conduct fitting tests on various regression equations according to Forest Measurement[1]， a textbook edited by Northeast Forestry University in 1985. 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. tomentosa 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 Ftest 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 Ftest with the reliability of 90%， the fitting equations for DBH 1 and its square， tree height， wood volume 2 and wood volume 3 all passed the Ftest with the reliability over 95%， and the fitting equations for others passed the Ftest and Rtest with the reliability over 99%， indicating that this mathematical model （the empirical fitting equation） was applicable as a whole. All 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 = 8.05a， that is， the current annual increment reached the peak when the tree reached 8 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 = 16.11 a， so the numerical maturity age of the trees was 16.11 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 indexes were doubled compared with the indexes， but the accuracies were equal to the Ftest values and Rtest 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 3 were obtained using the tendency chart of growth curves）， which could be used as the actual maturity ages of the tree wood. In addition to DBH 1， DBH 2， which showed great differences from their squares tm and tn， all other were very close to the values of tn. 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]. Conclusions and Applications
Different from previous studies， 3 different maturity ages were obtained， namely 11 years， 26 years and 41 years. Baased on the production practice， the 3 maturity ages were named as the maturity age for pulpwood （or short rotation）， maturity age for pillar wood （beam） and maturity age for peeler. The maturity age obtained from the fitting equation of DBH 1 was very close to the fitting value of wood volume 1， which was consistent with the results obtained from the previous traditional method. The maturity age obtained from the square of DBH 2 was very close to the fitting value obtained from the wood volume 2， which proved the preciseness of the research method. The test results were very close to the actual values. The division of the age groups was shown in Table 2.
The original standard was 9 years for pulpwood and 26 years for pillars， which was basically consistent with the conclusions of this study， but the standard for peeler has not yet been reported.
Discussion
The original maturity standard was 26 a， which was based on the standard set at the ageclass of 5 years. In this study， the ageclass 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 3 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 ageclass of 26 years （and above）. From this point of view， the issues concerning the maturity ages of trees cannot be the same. If conditions allow， some fitting equation tests can be conducted to find out more suitable maturity ages for the local so as to better guide the forestry production， which is of great importance. Due to the difficulty in collecting tree samples and limited funding， the parse wood materials were used， and the fitting empirical equation of tree growth was used to make up for the lack of the ages of parse wood， which also avoids the noise effects on the test results from the spatial differences and tree differentiation under various natural conditions. The maturity ages of the actual P. tomentosa forest are obtained in this way. After repeated verification， analysis and judgment， the empirical equation is applicable for the tree growth， but it is hard to make scientific explanation. Most of the observed effective figures in this study reach triple digits， which ensures the research accuracy objectively， so the method can be put into production practice for promotion and application. The improvement of the accuracy of observed data can make up for the insufficient ageclass data， so as to save the research expenditures. As for the accuracy of observation data， it should be improved in the subsequent scientific practice to ensure that the effective number of observation data reaches more than triple digits. 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.
Changgang WANG et al. Discussion on Individual Expected Maturity Age of Populus tomentosa Carr
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.
Editor： Na LI Proofreader： Xinxiu ZHU