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Abstract Through the study of parse wood materials， the fitting empirical equation of tree growth was obtained， a function with tree growth as 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 maturity age for the actual forest of Pinus thunbergii Parl was 51 a， and the ideal mature age was 66 a. And the application， research directions and precautions of the mature ages were proposed.
Key words Mature 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， the maturity ages of Pinus thunbergii Parl forest were studied.
Data source
The parse wood materials were collected from a dominant 43??year??old tree of P. thunbergii with normal growth from the south slope of Fengcheng Office of Haiyang， Yantai City in December， 2009. 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. Disks were intercepted a the tree height of 5 cm （disk 0）， 1.3， 3.6， 5.6 ， 7.6 and 8.6 m， and the disks were strictly interpreted in accordance with the technical requirements of parsing wood. Relevant information was collected with the age class of 5 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. thunbergii 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）， and 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 were established （Table 1）. As shown in Table 1， in addition to the fitting equations of ground diameter， the fitting equations for other items all passed the F??test and R??test with the reliability over 99.9%， 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. Wood volume 1 and 2 were far different from the actual conditions， so in this study， the fitting equation of wood volume 1 was used as a kind of actual state， while the fitting equation of wood volume 2 was used as a kind of ideal state. The maximum time of current annual increment and numerical mature age of trees by the fitting equations were illustrated with the ground diameter as an example. For the equation of growth rate of ground diameter （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=5.0 a， that is， the current annual increment reached the peak when the tree reached 5 years old or so， and the peak was a single one. For the equation of average growth rate of trees Y（t）/t （annual average increment， and only the extreme point was given in the paper）， the extreme point tm=9.9 a， so the numerical mature age of the tree was 9.9 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 directly in the paper. The numerical mature ages of each item were shown in Table 1. As shown in Table 1， the parameter values of DBH square were doubled compared with the values of DBH， 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， the fitting equation of DBH square was deliberately established. The average values of accumulated ideal mature age and actual mature age were close to the mature age of DBH square. 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， which could be used as the actual mature ages of the tree. In addition to DBH， which was very close to the tm values of the fitting equation， and wood volume， which was hard to determine because of the young age of the tree， the values of all other parameters were far different from the tm values. It was caused by the distortion of the relationship between the current annual growth and total average growth amount. However， the use of empirical equation could well solve such technical problems， and the research results were more reliable. Therefore， according to the research results and the needs for production practice， it was more suitable to set the mature age of accumulation volume as the mature age of the tree. Conclusion and Application
Through the analysis on the research results， the quantitative maturity of accumulation amount was used as the actual mature age of the tree. In order to facilitate the division of age groups， the division was not stepped over the age class. Thus， the actual mature age of P. thunbergii was 51a， and the ideal mature age was 66 a. The division of the age groups was shown in Table 2.
The results showed that the cutting （regenerating cutting） age of P. thunbergii stand should be 51-60 years， which was 10 years older than the original standard， rising by 25%， and it was determined by the traits of the tree itself. The ideal cutting （regenerating cutting） age of P. densiflora stand was 66-75 years， which was 25 older than the original standard， rising by 61%， and it was of great positive and practical significance for the improvement of the carbon fixation activities of trees.
Discussion
The original standard mature age was 26 a， which was lower than the mature age obtained in this study， so it is very important to analyze the simulation equations of actual forest stand. The main task of forestry production of pines in Shandong at this stage is to ensure the ecological benefits， while wood production is in a secondary position. Therefore， for ecological public??welfare forests that are mainly ecologically effective， the ideal forests that are less affected by external destructive noise can be upgraded and harvested after 66 years. This ideal mature age is recommended for the application in good site conditions like scenic forests， small damage from natural noise and social noise and the progenies of trees with long life. Compared with the other results in the same place， the actual maturity age is increased by 15 years， exceeding the original mature age standard and consistent with the ideal mature age， which can explain the practicability of the fitting equation. Due to the difficulty in collecting tree samples and the limited funds， empirical equation is used to fit the growth of parse wood materials to make up for the insufficient ages of parse wood， which also avoid the noise effects of the space??time differences of natural conditions and tree differentiation on the test results. The obtained actual mature age of P. thunbergii forest stand has been verified repeatedly. The analysis and judgment on the results show that the empirical equation does fit the growth of trees， but it is hard to make scientific explanations. Due to limited time and capability， various deviations are inevitable， which can only be improved and developed in the future production research practice. The proposed forestry production proposal only represents personal opinions. After all， it is obtained by analyzing the parse wood materials of an individual tree， which is young in age. The obtained conclusion is inevitably biased， and can only be applied after being approved by relevant experts and tested by production practice.
References
[1] LANG KJ. Forest measurement[M]. Northeast Forestry University， 1985： 283-296.
[2] CHEN HH. 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 Mature 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， the maturity ages of Pinus thunbergii Parl forest were studied.
Data source
The parse wood materials were collected from a dominant 43??year??old tree of P. thunbergii with normal growth from the south slope of Fengcheng Office of Haiyang， Yantai City in December， 2009. 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. Disks were intercepted a the tree height of 5 cm （disk 0）， 1.3， 3.6， 5.6 ， 7.6 and 8.6 m， and the disks were strictly interpreted in accordance with the technical requirements of parsing wood. Relevant information was collected with the age class of 5 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. thunbergii 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）， and 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 were established （Table 1）. As shown in Table 1， in addition to the fitting equations of ground diameter， the fitting equations for other items all passed the F??test and R??test with the reliability over 99.9%， 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. Wood volume 1 and 2 were far different from the actual conditions， so in this study， the fitting equation of wood volume 1 was used as a kind of actual state， while the fitting equation of wood volume 2 was used as a kind of ideal state. The maximum time of current annual increment and numerical mature age of trees by the fitting equations were illustrated with the ground diameter as an example. For the equation of growth rate of ground diameter （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=5.0 a， that is， the current annual increment reached the peak when the tree reached 5 years old or so， and the peak was a single one. For the equation of average growth rate of trees Y（t）/t （annual average increment， and only the extreme point was given in the paper）， the extreme point tm=9.9 a， so the numerical mature age of the tree was 9.9 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 directly in the paper. The numerical mature ages of each item were shown in Table 1. As shown in Table 1， the parameter values of DBH square were doubled compared with the values of DBH， 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， the fitting equation of DBH square was deliberately established. The average values of accumulated ideal mature age and actual mature age were close to the mature age of DBH square. 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， which could be used as the actual mature ages of the tree. In addition to DBH， which was very close to the tm values of the fitting equation， and wood volume， which was hard to determine because of the young age of the tree， the values of all other parameters were far different from the tm values. It was caused by the distortion of the relationship between the current annual growth and total average growth amount. However， the use of empirical equation could well solve such technical problems， and the research results were more reliable. Therefore， according to the research results and the needs for production practice， it was more suitable to set the mature age of accumulation volume as the mature age of the tree. Conclusion and Application
Through the analysis on the research results， the quantitative maturity of accumulation amount was used as the actual mature age of the tree. In order to facilitate the division of age groups， the division was not stepped over the age class. Thus， the actual mature age of P. thunbergii was 51a， and the ideal mature age was 66 a. The division of the age groups was shown in Table 2.
The results showed that the cutting （regenerating cutting） age of P. thunbergii stand should be 51-60 years， which was 10 years older than the original standard， rising by 25%， and it was determined by the traits of the tree itself. The ideal cutting （regenerating cutting） age of P. densiflora stand was 66-75 years， which was 25 older than the original standard， rising by 61%， and it was of great positive and practical significance for the improvement of the carbon fixation activities of trees.
Discussion
The original standard mature age was 26 a， which was lower than the mature age obtained in this study， so it is very important to analyze the simulation equations of actual forest stand. The main task of forestry production of pines in Shandong at this stage is to ensure the ecological benefits， while wood production is in a secondary position. Therefore， for ecological public??welfare forests that are mainly ecologically effective， the ideal forests that are less affected by external destructive noise can be upgraded and harvested after 66 years. This ideal mature age is recommended for the application in good site conditions like scenic forests， small damage from natural noise and social noise and the progenies of trees with long life. Compared with the other results in the same place， the actual maturity age is increased by 15 years， exceeding the original mature age standard and consistent with the ideal mature age， which can explain the practicability of the fitting equation. Due to the difficulty in collecting tree samples and the limited funds， empirical equation is used to fit the growth of parse wood materials to make up for the insufficient ages of parse wood， which also avoid the noise effects of the space??time differences of natural conditions and tree differentiation on the test results. The obtained actual mature age of P. thunbergii forest stand has been verified repeatedly. The analysis and judgment on the results show that the empirical equation does fit the growth of trees， but it is hard to make scientific explanations. Due to limited time and capability， various deviations are inevitable， which can only be improved and developed in the future production research practice. The proposed forestry production proposal only represents personal opinions. After all， it is obtained by analyzing the parse wood materials of an individual tree， which is young in age. The obtained conclusion is inevitably biased， and can only be applied after being approved by relevant experts and tested by production practice.
References
[1] LANG KJ. Forest measurement[M]. Northeast Forestry University， 1985： 283-296.
[2] CHEN HH. 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.