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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. The mature age of tree growth was obtained through mathematical operations such as function derivation. The obtained expected maturity ages for the actual forests of Robinia pseudoacacia were 36， 46 and 56 a， respectively， which could be the mature ages for commercial forest， protection forest and special??purpose forest. 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 mature 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 mature ages of Robinia pseudoacacia forest were studied. R. pseudoacacia is an indigenous tree species introduced from the foreign countries and adapt to the local environment， and it has become the major tree species in the upland afforestation in Shandong Province. Since its roots can fix nitrogen， showing strong vitality， R. pseudoacacia has become the major tree species for pitwood. And its tender leaves and branches as well as dry leaves are good feed sources. However， in production practice， the regeneration of sprout tillers has led to germplasm degeneration， and the stumps are hard to remove， making people treat it as a harmful species. These problems can be solved through afforestation by seedlings and using appropriate methods （such as peeling off the skin in the peak season of growth）. It is of important significance to explore the mature ages of R. pseudoacacia.
Data source
Due to limited funding， the previous survey materials were used in the paper. The parse wood materials were collected from a 34??year??old tree of R. pseudoacacia with normal growth from Dafengshan Forest Farm in Changqing District of Jinan City on November 11， 1983. 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 at the tree height of 5 cm （disk 0）， 1.3 ， 3.6 ， 5.6 ， 7.6 ， 9.6 and 10.6 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 2 years. 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 factors with the greatest impact on R. pseudoacacia are 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 and wood volumes 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， the fitting equations for ground diameter 1， ground diameter 3， DBH 2??4 and their squares， tree height and wood volumes passed the F??test with the reliability of 90%， fitting equations for ground diameter 2， ground diameter 4 and wood volume 7 passed with the reliability of 95%， and the others 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. 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 growth rate equation 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=6.00 a， that is， the current annual increment reached the peak when the tree reached 6 years old， and the peak was a single one. For the equation of average tree growth rate Y（t）/t （annual average increment， and only the extreme point was given in the paper）， the extreme point tm=11.99 a， so the numerical maturity age of the tree was 11.99 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 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， such fitting equation was deliberately established in this study. The fitting results of DBH 2 square， wood volumes 2 and 5 were very close， and the fitting results of DHB 3 square， wood volumes 3 and 6 were close to each other. 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 （those for the trees with ages over 34 a were obtained from the trend chart of growth curve）， which could be used as the actual mature ages of the tree. The fitting results showed that the fitted values tm and tn were close to each other. However， the fitted values from the equations from wood volume were close to the tn values of DBH （DBH square）， so the values were set as the mature age of wood volume. The maturity of a tree is mainly concerned with the maturity of wood volume， so according to the research results and the needs in production practice， it was more suitable to set the mature age of accumulation volume as the mature age of the tree. Conclusion and Application
According to the traditional methods， the mature age of DBH was 26 a， which was consistent with the result of 25.72 a from the fitting equation of DBH1. In actual production practice， setting the mature age of R. pseudoacacia to 26 a was in accordance with the traditional meaning， and consistent with the national standard of China. However， the analysis on the results showed that the actual mature ages for R. pseudoacacia were 36， 46 and 56 a using the mature age of accumulation volume as the standard， which could be the mature ages for commercial forest， protection forest and special??purpose forest of such tree species. The division of the age groups was shown in Table 2.
According to the analysis on the fitting equation of wood volume 4， the sample tree could have another growth peak in 113 years， indicating that the lifetime of the tree could reach up to 150a. Therefore， this study is of practical guiding significance to predict the lifetime of trees.
Discussion
The original standard mature age is 26 a， which is based on the standard set at the age class of 5 years. In this study， the mature age is 36 a， which is based on the age class of 2 years. There is a 10??year difference between the 2. The original standard spares no consideration to the fact that through variety selection and improvement， R. pseudoacacia has got great improvement in growth rate， and has become one of the dominant fast growing tree species in Shandong. With the decrease of age class， the results are more close to the reality. With time going by， it should make appropriate adjustment according to the actual local conditions with the times. As an indigenous afforestation tree species， R. pseudoacacia has been widely applied in the barren mountain afforestation in Shandong， so it is of positive practical significance to explore the growth rules of R. pseudoacacia. With no available observation data of the 36 age class （and above）， the mature age of the tree was obtained through equation fitting based on the analysis of growth curves. From this point of view， the issue of maturity of trees cannot be uniform and unchanging. If conditions permit， it is important to do some fitting equation tests to find a more suitable mature age for local guidance in order to better guide forestry production. 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 R. pseudoacacia 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 mature 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 mature ages of Robinia pseudoacacia forest were studied. R. pseudoacacia is an indigenous tree species introduced from the foreign countries and adapt to the local environment， and it has become the major tree species in the upland afforestation in Shandong Province. Since its roots can fix nitrogen， showing strong vitality， R. pseudoacacia has become the major tree species for pitwood. And its tender leaves and branches as well as dry leaves are good feed sources. However， in production practice， the regeneration of sprout tillers has led to germplasm degeneration， and the stumps are hard to remove， making people treat it as a harmful species. These problems can be solved through afforestation by seedlings and using appropriate methods （such as peeling off the skin in the peak season of growth）. It is of important significance to explore the mature ages of R. pseudoacacia.
Data source
Due to limited funding， the previous survey materials were used in the paper. The parse wood materials were collected from a 34??year??old tree of R. pseudoacacia with normal growth from Dafengshan Forest Farm in Changqing District of Jinan City on November 11， 1983. 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 at the tree height of 5 cm （disk 0）， 1.3 ， 3.6 ， 5.6 ， 7.6 ， 9.6 and 10.6 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 2 years. 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 factors with the greatest impact on R. pseudoacacia are 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 and wood volumes 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， the fitting equations for ground diameter 1， ground diameter 3， DBH 2??4 and their squares， tree height and wood volumes passed the F??test with the reliability of 90%， fitting equations for ground diameter 2， ground diameter 4 and wood volume 7 passed with the reliability of 95%， and the others 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. 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 growth rate equation 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=6.00 a， that is， the current annual increment reached the peak when the tree reached 6 years old， and the peak was a single one. For the equation of average tree growth rate Y（t）/t （annual average increment， and only the extreme point was given in the paper）， the extreme point tm=11.99 a， so the numerical maturity age of the tree was 11.99 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 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， such fitting equation was deliberately established in this study. The fitting results of DBH 2 square， wood volumes 2 and 5 were very close， and the fitting results of DHB 3 square， wood volumes 3 and 6 were close to each other. 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 （those for the trees with ages over 34 a were obtained from the trend chart of growth curve）， which could be used as the actual mature ages of the tree. The fitting results showed that the fitted values tm and tn were close to each other. However， the fitted values from the equations from wood volume were close to the tn values of DBH （DBH square）， so the values were set as the mature age of wood volume. The maturity of a tree is mainly concerned with the maturity of wood volume， so according to the research results and the needs in production practice， it was more suitable to set the mature age of accumulation volume as the mature age of the tree. Conclusion and Application
According to the traditional methods， the mature age of DBH was 26 a， which was consistent with the result of 25.72 a from the fitting equation of DBH1. In actual production practice， setting the mature age of R. pseudoacacia to 26 a was in accordance with the traditional meaning， and consistent with the national standard of China. However， the analysis on the results showed that the actual mature ages for R. pseudoacacia were 36， 46 and 56 a using the mature age of accumulation volume as the standard， which could be the mature ages for commercial forest， protection forest and special??purpose forest of such tree species. The division of the age groups was shown in Table 2.
According to the analysis on the fitting equation of wood volume 4， the sample tree could have another growth peak in 113 years， indicating that the lifetime of the tree could reach up to 150a. Therefore， this study is of practical guiding significance to predict the lifetime of trees.
Discussion
The original standard mature age is 26 a， which is based on the standard set at the age class of 5 years. In this study， the mature age is 36 a， which is based on the age class of 2 years. There is a 10??year difference between the 2. The original standard spares no consideration to the fact that through variety selection and improvement， R. pseudoacacia has got great improvement in growth rate， and has become one of the dominant fast growing tree species in Shandong. With the decrease of age class， the results are more close to the reality. With time going by， it should make appropriate adjustment according to the actual local conditions with the times. As an indigenous afforestation tree species， R. pseudoacacia has been widely applied in the barren mountain afforestation in Shandong， so it is of positive practical significance to explore the growth rules of R. pseudoacacia. With no available observation data of the 36 age class （and above）， the mature age of the tree was obtained through equation fitting based on the analysis of growth curves. From this point of view， the issue of maturity of trees cannot be uniform and unchanging. If conditions permit， it is important to do some fitting equation tests to find a more suitable mature age for local guidance in order to better guide forestry production. 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 R. pseudoacacia 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.