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Abstract Based on 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， and the obtained expected mature age for Larix kaempferi was 46 years. 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 age of Larix kaempferi was studied. L. kaempferi is one of the major tree species that have been introduced to Shandong Province for the greening and afforestation of barren hills， and its growth speed is superior to other coniferous species. The vigorous construction of L. kaempferi forest is of great and profound significance to improve the proportional structure of softwood pulp and hardwood pulp. Therefore， it is of great significance to explore the mature age of L. kaempferi.
Data Sources
L. kaempferi resources are quite scarce， and most of them are introduced after the 1950s. Due to limited funding， the previous survey materials were used in the paper. The parse wood materials were collected from a dominant 24 year old L. kaempferi tree with normal growth from Laoshan Forest Farm of Qingdao City on September 17， 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. 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， 13.6 and 14.6 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 2 years.
Research Methods
In order to save research costs， based on the analysis of parse wood data， fitting tests were conducted to various regression equations using previous research[1-3] by referring to previous research methods and processes[4-5] and research results[6]. Finally， the following mixed empirical equation was adopted to study the growth of the tree： y（t） =e a-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 growth of L. kaempferi was greatly affected by the precipitation volume and uniformity of spatial and temporal distribution. Based on the empirical equations to fit the process of tree growth， the quantitative 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 same method was used to get the quantitative mature ages for the growth of DBH， DBH square， tree height， wood volume.
Research Process
Based on years of practical experience， comparative tests were carried out to the application of unit dimension. The results showed that the smaller the dimension （unit）， the closer it was to the actual situation， the higher the accuracy and reliability of the test （the dimension will not be in line with the actual situation if it is small to a certain extent. Mathematical methods can be used to calculate the best dimension suitable for the actual situation， but for the sake of easy understanding， the decimal system is adopted， that is， the dimensional change can only be the geometrical progression of 10， which can achieve the accuracy required by the experiment）. However， considering the limitation of test conditions， it would be better to measure the observed data accurately to 0.1 mm in all aspects， because smaller dimensions require more precise instruments to increase the accuracy， which would increase the working costs by hundreds of times to achieve research accuracy and coincide with the actual situation. On the other hand， as long as the significant figures reached the correlated digits， the effect would be small on the test results. Although there may be slight deviation， no impact would be done to the expression of the true value.
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 tree height and wood volume 1， which passed the F test with the reliability over 90%， the fitting equations for other items all passed the F test and R test with the reliability of 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 quantitative mature age of trees from the fitting equations were illustrated with DBH as an example. For the growth rate equation of DBH （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 t z=5.24 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 t m=10.57 a， so the quantitative mature age of the tree was 10.57 a. In this paper， only the fitting equations for DBH 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 meanings of symbols of t z， t m. Therefore， the calculation results were given directly in the paper. The quantitative mature ages of each item were shown in Table 1. As shown in Table 1， the parameter values were doubled compared with the values of the indicators and the squares of the indicators， but the accuracy was equal to the F test values and R test values. This was caused by the exponential mathematical relationship. This fitting equation was established specifically to compare with the accumulation fitting equation. In Table 1， the values of t n were the ages at the intersection points of the curves of current annual increments and annual average increments of the sample wood （wood volume was obtained from the tendency chart of growth curve）， which could be used as the actual mature age of the tree. In addition to DBH， which had the values twice of t m， all other items had close values to t m. Therefore， the empirical equation was more applicable for wood volume， and the results were more reliable. Concerning with the maturity of trees， the major factor was the maturity of wood volume， and thus， 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. 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. Conclusion and Application
Through the analysis on the research results， the quantitative maturity of accumulation volume was used as the actual mature age of the tree， which was 47 years for L. kaempferi. The division of the age groups was shown in Table 2.
Discussion
The original standard mature age was 41a， mainly referring to the national standard， and it was made using a 10 year age class standard. On the other hand， in this study， the research conclusion was made using 2 years as the age class， which is much closer to the actual situations. Therefore， with the passage of time， appropriate adjustments should be made according to actual local conditions. Moreover， as an introduced tree species， L. kaempferi has been widely used in the greening of barren mountains in Shandong Province， and its development is of great significance to improve the proportional structure of softwood pulp and hardwood pulp， so it is very necessary to explore its growth law. In this study， the dimension has been made flexible use of to solve the problem， and the applicability of dimension changes have been pointed out. Especially， with the volume decreasing， the value of energy parameter a and fitting accuracy of the equation remain unchanged， only the value of b， which represents the mature age， changes regularly （Table 1 only gives the fitting equation with higher test accuracy which are closer to the reality， the others omit ）. In this way， it makes it possible to obtain the mature age of trees through equation fitting based on the analysis and judgment of growth curve tendency when there is a lack of observation data of tree age class of 24-46 years and above. Thus， the maturity of trees can not remain unchanged， and if conditions permit， some fitting equation tests can be made to find out the mature age which is more suitable for the local situation， so as to better guide forestry production， which is very important. In this study， 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 L. kaempferi 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. The limitation of this study lies in the lack of observation data of high age class， funding and time， and it is impossible to make the length unit accurate to micrometer， so there are some uncertainties. To solve these problems， it is necessary to increase the investment of human and material resources， and it is hoped that interested colleagues can set up more detailed and scientific research topics. On the other hand， most of the observed effective figures in this study have reached more than three digits， which objectively guarantees the research accuracy and can be promoted and applied in the practice of real forestry production. The improvement of the accuracy of the observation data can make up for the shortage of the age class data to save the research expenses. As for the accuracy of the observed data， it should be improved in the future scientific practice to ensure that the significant digits of observation data reach more than three. As for the reasonable adjustment of the dimension， the results of this study show striking similarities to another study on another parse wood tree （Quercus acutissima from Pingdu City）， so it has certain reference significance in the future practice of scientific research. 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 suggestions only represent personal opinions. After all， the conclusion is obtained by analyzing the parse wood materials of an individual tree， and can only be applied after being approved by relevant experts and tested by production practice.
Agricultural Biotechnology2019
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 age of Larix kaempferi was studied. L. kaempferi is one of the major tree species that have been introduced to Shandong Province for the greening and afforestation of barren hills， and its growth speed is superior to other coniferous species. The vigorous construction of L. kaempferi forest is of great and profound significance to improve the proportional structure of softwood pulp and hardwood pulp. Therefore， it is of great significance to explore the mature age of L. kaempferi.
Data Sources
L. kaempferi resources are quite scarce， and most of them are introduced after the 1950s. Due to limited funding， the previous survey materials were used in the paper. The parse wood materials were collected from a dominant 24 year old L. kaempferi tree with normal growth from Laoshan Forest Farm of Qingdao City on September 17， 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. 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， 13.6 and 14.6 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 2 years.
Research Methods
In order to save research costs， based on the analysis of parse wood data， fitting tests were conducted to various regression equations using previous research[1-3] by referring to previous research methods and processes[4-5] and research results[6]. Finally， the following mixed empirical equation was adopted to study the growth of the tree： y（t） =e a-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 growth of L. kaempferi was greatly affected by the precipitation volume and uniformity of spatial and temporal distribution. Based on the empirical equations to fit the process of tree growth， the quantitative 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 same method was used to get the quantitative mature ages for the growth of DBH， DBH square， tree height， wood volume.
Research Process
Based on years of practical experience， comparative tests were carried out to the application of unit dimension. The results showed that the smaller the dimension （unit）， the closer it was to the actual situation， the higher the accuracy and reliability of the test （the dimension will not be in line with the actual situation if it is small to a certain extent. Mathematical methods can be used to calculate the best dimension suitable for the actual situation， but for the sake of easy understanding， the decimal system is adopted， that is， the dimensional change can only be the geometrical progression of 10， which can achieve the accuracy required by the experiment）. However， considering the limitation of test conditions， it would be better to measure the observed data accurately to 0.1 mm in all aspects， because smaller dimensions require more precise instruments to increase the accuracy， which would increase the working costs by hundreds of times to achieve research accuracy and coincide with the actual situation. On the other hand， as long as the significant figures reached the correlated digits， the effect would be small on the test results. Although there may be slight deviation， no impact would be done to the expression of the true value.
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 tree height and wood volume 1， which passed the F test with the reliability over 90%， the fitting equations for other items all passed the F test and R test with the reliability of 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 quantitative mature age of trees from the fitting equations were illustrated with DBH as an example. For the growth rate equation of DBH （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 t z=5.24 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 t m=10.57 a， so the quantitative mature age of the tree was 10.57 a. In this paper， only the fitting equations for DBH 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 meanings of symbols of t z， t m. Therefore， the calculation results were given directly in the paper. The quantitative mature ages of each item were shown in Table 1. As shown in Table 1， the parameter values were doubled compared with the values of the indicators and the squares of the indicators， but the accuracy was equal to the F test values and R test values. This was caused by the exponential mathematical relationship. This fitting equation was established specifically to compare with the accumulation fitting equation. In Table 1， the values of t n were the ages at the intersection points of the curves of current annual increments and annual average increments of the sample wood （wood volume was obtained from the tendency chart of growth curve）， which could be used as the actual mature age of the tree. In addition to DBH， which had the values twice of t m， all other items had close values to t m. Therefore， the empirical equation was more applicable for wood volume， and the results were more reliable. Concerning with the maturity of trees， the major factor was the maturity of wood volume， and thus， 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. 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. Conclusion and Application
Through the analysis on the research results， the quantitative maturity of accumulation volume was used as the actual mature age of the tree， which was 47 years for L. kaempferi. The division of the age groups was shown in Table 2.
Discussion
The original standard mature age was 41a， mainly referring to the national standard， and it was made using a 10 year age class standard. On the other hand， in this study， the research conclusion was made using 2 years as the age class， which is much closer to the actual situations. Therefore， with the passage of time， appropriate adjustments should be made according to actual local conditions. Moreover， as an introduced tree species， L. kaempferi has been widely used in the greening of barren mountains in Shandong Province， and its development is of great significance to improve the proportional structure of softwood pulp and hardwood pulp， so it is very necessary to explore its growth law. In this study， the dimension has been made flexible use of to solve the problem， and the applicability of dimension changes have been pointed out. Especially， with the volume decreasing， the value of energy parameter a and fitting accuracy of the equation remain unchanged， only the value of b， which represents the mature age， changes regularly （Table 1 only gives the fitting equation with higher test accuracy which are closer to the reality， the others omit ）. In this way， it makes it possible to obtain the mature age of trees through equation fitting based on the analysis and judgment of growth curve tendency when there is a lack of observation data of tree age class of 24-46 years and above. Thus， the maturity of trees can not remain unchanged， and if conditions permit， some fitting equation tests can be made to find out the mature age which is more suitable for the local situation， so as to better guide forestry production， which is very important. In this study， 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 L. kaempferi 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. The limitation of this study lies in the lack of observation data of high age class， funding and time， and it is impossible to make the length unit accurate to micrometer， so there are some uncertainties. To solve these problems， it is necessary to increase the investment of human and material resources， and it is hoped that interested colleagues can set up more detailed and scientific research topics. On the other hand， most of the observed effective figures in this study have reached more than three digits， which objectively guarantees the research accuracy and can be promoted and applied in the practice of real forestry production. The improvement of the accuracy of the observation data can make up for the shortage of the age class data to save the research expenses. As for the accuracy of the observed data， it should be improved in the future scientific practice to ensure that the significant digits of observation data reach more than three. As for the reasonable adjustment of the dimension， the results of this study show striking similarities to another study on another parse wood tree （Quercus acutissima from Pingdu City）， so it has certain reference significance in the future practice of scientific research. 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 suggestions only represent personal opinions. After all， the conclusion is obtained by analyzing the parse wood materials of an individual tree， and can only be applied after being approved by relevant experts and tested by production practice.
Agricultural Biotechnology2019
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.