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Abstract Through the study on data of analytic tree， the fitting empirical equation for tree growth was obtained， i.e.， the function with increment as variable and annual precipitation as the independent variable. The threshold value of annual precipitation for tree growth was obtained through mathematical operation including derivation. It was concluded that Larix gmelinii grows the fast under the annual precipitation of about 545.1 mm， and negative growth would occur if the annual precipitation is lower than 349 mm and higher than 1 132.8 mm. Furthermore， the application value， research direction and matters needing attention were pointed out.
Key words Threshold； Limit value； Empirical equation； Analytic tree； Annual precipitation
In various production practices of planting industry， waterlogging is a common problem， that is to say the uneven spatial and temporal distribution of precipitation would result in adverse effect on production， especially in forestry production. Many scholars concluded that the change of annual precipitation has no significant effect on tree growth according to growth ring information and annual precipitation information. The writers deem that the conclusion is not accurate， and might be due to inaccurate measurement （due to false annual ring）， or noncorrespondence between precipitation and local information， or the comprehensive effect of other noises. Also， some scholars concluded that precipitation is positively or negatively correlated with tree growth from local information， and some scholars reckoned the precipitation of the previous year has a more significant effect on tree growth in next year. In this study， the relationship between tree growth and annual precipitation was studied using the data of analytic tree taking Larix gmelinii as an example， so as to verify above viewpoints.
Data Source
The precipitation information over the years was provided by the meteorological department. The analytic tree information was collected from a 24yearold L.gmelinii growing normally in Laoshan forest farm in Laoshan district， Qingdao City on September 17， 1983. The breast height section was segmented according to 2.6 m， and other sections were segmented according to 2 m. Discs were cut at the tree heights of 5 cm （0 plate）， 1.3， 3.6， 5.6， 7.6， 9.6， 11.6， 13.6 and 14.6 m， respectively， and interpretation was performed strictly according to the technical points of analytic tree. Related information was collected with two years as an age class. Research Method
In order to save research cost， fitting tests were carried out to various regression equations using tree growth empirical equations according to "Forest Mensuration" and "Mathematical Statistics" edited by Northeast Forestry University， and with reference to the research results of Wang et al.[5-11]， and finally， the mixed type empirical equation， y（t）=ea-b/t （a and b are tobedetermined exponential parameters， and e is the base number of natural logarithm： 2.718 28……）， was selected for the study of tree growth process. Because tree growth is influenced by various factors， and the amount and spatial and temporal distribution of precipitation has the highest effect on L. gmelinii， the writers tried to fit the tree growth process using empirical equations， and through the acquisition of the accumulated annual precipitation according to DBH fitting equations （including the equation obtained in derivation， as described in the research process）， various precipitation indices affecting growth of treemeasuring factors including the diameter at breast height was obtained. The threshold value of annual precipitation for tree growth was calculated using y（p）=a-b/p， y（p）=a+bp and y（p）=a+blnp， and the threshold value of annual precipitation limiting tree growth was calculated using y（p）=1/（a-blnp） and y（p）=1/（a-bp）.
Research Process
Through the check analysis of tree growth equation fitting within two years， three treemeasuring factors （tree height， DBH and volume of timber） all had fitting relation with precipitation. Unary linear equations （direction application of unary linear equations） were formed by various arithmetical operations on various fitting equations. Then， parameters a and b were obtained by unary linear regression， and F test and correlation coefficient （R） test were carried out. If the tests are passed， a tree growth fitting equation is established， and the calculation results are shown in Table 1. Table lookup showed that the tests were passed with a reliability over 90%， indicating that these mathematical models （i.e.， empirical fitting equations） are applicative overall. The problem of solving the maximum current annual increment time and quantitative maturity of trees using fitting equations was illustrated taking the DBH item as an example. The precipitation fitting mixed equation of tree DBH growth y（p）=ea-b/p （the increment changing with precipitation， obtained by derivation of Y（t） function， the process of which was omitted， only the extreme point was given， the same below） showed the extreme value pz=272.6 mm， i.e.， when the accumulated precipitation reaches 272.6 mm in two years， the current water resourcecontributed increment reaches its peak （i.e.， the instant use efficiency of water is the highest）， which is the only peak； and the average water use efficiency equation of average tree growth Y（p）/p （precipitation converted from water resource consumed by annual average unit increment， the calculation process of which was omitted， only the extreme value was given， the same as below） had the extreme value pm=545.1 mm， i.e.， when the accumulated precipitation reached 545.1 mm in two years， the current average water resource use efficiency was the highest. Only the calculation of DBH fitting equation was described above， other fitting equations were the same as it， and detailed description was omitted. In the discussion below， growth fitting equations， and symbols such as pz and pm， also had the same meaning， and the calculation results were given without repeated description and explanation. The various calculated parameters and various threshold values are shown in Table 1. It could be seen from Table 1 that in the equation y（p）=ea-b/p， compared with DBH， various parameters of the square of DBH were improved by one time， their precision， F values and R values were equal， which was caused by the exponent mathematic relation， and in order to compared with accumulation， a fitting equation was also established with the square of DBH as a treemeasuring factor. Various equations all could pass F test and R test， and the precision is shown in Table with a lowest value of 90%， and the highest value of 99.9%. For ease of listing， symbols were used， for instance， 139.8 mm （±2 a） refers to that within two years， if the annual average precipitation is lower than 139.8 mm， positive growth turns to negative growth， and this value is called threshold value； 1 974.23 mm （±∞2 a） means that within two years， when the effective annual average precipitation reaches 1 974.23 mm， theoretical unlimited growth， the maximum increment is reached， but once the value is exceeded， immoderately destructive negative growth occurred， so the value is also known as threshold value （the same below）.
It could be seen from the test results that the change in precipitation had greater effect on the change of volume of timber before 16 a， the water demand was larger in early period， and water was the main limiting condition for tree growth； and in 13-15 a which was the peak period of tree growth， precipitation had greater effect on tree height； and in 6-24 a， precipitation had greater effect on the growth of DBH.
It could be seen from the test results that tree height exhibited negative correlation， and DBH and volume of timber showed periodic positive and negative correlation， and thus had no close relation to the change of annual precipitation.
It could be seen from Table 1 that equations V5 and V6 showed more proximate threshold values， with a relative error limit only of 1.82%， the test precision reached 98% and 99.9%， respectively， and therefore， these test results are more reliable. Equations V1 and V4 exhibited more proximate threshold values， with a relative error limit only of 0.44%. According to the fitted values with higher precision， the threshold value for tree growth was annual precipitation from 461.7 to 1 044.7 mm， exceeding which negative growth of trees， even daeath， might be caused. The equation of D21.3 also had the test precision of 99%， and D21.3 was in nearlinear correlation with the volume of timber. Through the derivation operation of the equation， the threshold value pz=545.1 mm was obtained， indicating that within two years， if the annual average accumulation reaches this value， the current growth rate of D21.3 is the highest； and the other threshold value pm=1 090.2 mm was also obtained， indicating that within two years， if the annual average accumulation reaches this value， the annual average growth rate is the highest. Obviously， the Pm value exceeded the threshold value 1 044.7 mm， but the two values were proximate， so the value with higher precision was chosen， the threshold value for tree growth was corrected to be from 461.7 to 1 090.2 mm. Through the check analysis on fitting of the growth process equations of the main treemeasuring factors within 24 years， all the treemeasuring factors all showed equation fitting relation， with very high reliability and correlation coefficient. The precision was about 99% mostly， nearly approximate to 100%. Various equation relation and values are shown in Table 2 （D1.3， H and V represents DBH， tree height and volume of timber， respectively， the same below）. The meanings of the values were the same as those of the DBH fitting equations within two years. Some of the precision in Table 2 is 100%， which means that the value is infinitely approximate to 100%， while the correlation coefficient is exactly 100% when the value is 1. However， due to the influence of subjective and objective noise， the fitting precision hardly could reach 100% （the same below）， and a higher fitting precision indicates the accuracy of the data and the applicability of the selected equation. Therefore， the test results in this study have higher scientific value. It could be seen from Table 2 that all the treemeasuring factors were in positive correlation with precipitation， and it was speculated that the periodic alternation of positive and negative correlation in individual years might be related to the unstable yielding phenomenon in seed propagation of trees. The growth and development of L. gmelinii is mainly vegetative growth， and therefore， the fitting equations of accumulated growth exhibited positive correlation in the whole process.
It could be seen from Table 2 that equations V1 and V2， and V3 and V4 showed more approximate threshold values， with relative error limits of 1.46% and 1.94%， respectively， and the test precision was approximate to 100%. Therefore， this group of test results were more reliable. The fitted values with higher precision were selected， and the threshold value of precipitation for tree growth was from 349.9 to 1 132.8 mm， beyond which negative growth of trees， even death， might be caused. The equation of D21.3 also had precision approximate to 100%， and D21.3 was in nearlinear correlation with the volume of timber. Through the derivation operation of the equation， one extreme point was obtained as pz=700.3 mm， which meant that when the average annual accumulation of precipitation reaches this value within 24 a， the current growth rate of D21.3 would be the highest； and the other extreme point was obtained as pm=1 400.6 mm， which meant that when the average annual accumulation of precipitation reaches this value within 24 a， the average annual growth rate would the highest. Obviously， the Pm value exceeded 1 132.8 mm， but the two values were close， so the value with higher precision was chosen， i.e.， the limit value of tree growth was corrected as from 349.9 to 1 132.8 mm. If irrigation is allowed， it is necessary to irrigate the trees when the annual precipitation is lower than 349.9 mm， mainly in the modes of sprinkling irrigation and drop irrigation. The optimal irrigation amount is equivalent to the annual precipitation of 700.3 mm （including natural precipitation， the same blow）， and the largest amount should not exceed 1 400.6 mm， beyond which the problem of drainage would be caused. The fitted vale of the equation H7 was extremely approximate to the pm value obtained from D1.3=exp（a-b/p）， and the two values were nearly the same， indicating very high test precision. The fitted values of equations V7 and V8 were far different from the actual conditions， but showed the trend that the water demand for tree growth increased with the age increasing. The calculation results of the materials within 24 a including the fitting results of the materials within 2 a exhibited improved test precision， and confirmed each other.
Through check analysis， when comparing current precipitation with the precipitation of the previous year， the F value was 1.333 （2.32 was the F value under the reliability of 90%）； and within 24 years， when comparing the accumulated value of d1.3 of the previous year with the fitted value of the equation d1.3=exp（a-b/p） of the current year and the actual value of another sample tree， the F values were 1.206 and 1.141， respectively， indicating there were no significant differences， and therefore， not only the fitting equation was established， but also the effects of current precipitation and the precipitation of the previous year on tree were hard to be differentiated. The data of the previous years were also subjected to fitting analysis， and except that the test precision of the previous year was slightly higher than the current year， no big differences were found. Therefore， repeated description was omitted. Some scholars deem that there is a lack of basis for the speculation that the precipitation of the previous year has a more significant effect on tree growth in the current year， and the fact might be that the precipitation of the previous year would affect the reproduction growth of trees， and then indirectly affects the vegetative growth in the current year. However， the writers deem that this is one of the reasons for the periodic change of the positive and negative correlation with annual precipitation， but the effect is very little， and the main reason is the periodic development law of trees. Other equations were also subjected to suitability test by the same method， and the fitted values of the equations all had no significant differences from the indices of another sample tree. However， there was a significant difference between the P. thunbergii×P. densiflora at another location and the P. thunbergii at the same location. Therefore， different tree species could not use the same equation， because tree species differ in growth and development laws.
Conclusions and Application
The results of this study showed that various treemeasuring factors had very remarkably correlation with the amount of precipitation and its distribution. For the growth of volume of timber， under the condition of uniform spatial and temporal precipitation distribution， when the average annual precipitation reached 700.3 mm， the annual increment reached the maximum value， and the instant water use efficiency was the highest； and when the average annual precipitation reached 1 400.6 mm， the average annual increment contributed by unit water resource reached its maximum value， and the average water use efficiency was the highest. Within 24 a， when the currently precipitation was lower than 349.9 mm， trees grew negatively and even died； and when the average annual precipitation reached 1 400.6 mm， the volume of timber had an infinite increment theoretically， but once the threshold value was exceeded， destructive negative growth of trees occurred， i.e.， death of trees due to waterlogging （the threshold value of DBH was 1 457.9mm， and that of tree height was 1 134.9 mm）. The fitted values of volume of timber and tree height confirmed each other， indicating that the test was very accurate. Within 2 a， when the average annual precipitation reached 545.1 mm， the instant use efficiency of water resource was the highest； and when that annual average precipitation reached 1 090.2 mm， the average annual use efficiency of water resource was the highest. According to above results， it is advised to irrigate the tree properly in the mode of drop irrigation when the forecasted annual precipitation is lower than 545.1 mm， and the largest amount of irrigation should not exceed 1 090.2 mm （including natural precipitation）. When the average annual precipitation exceeds 1 090.2 mm， drainage and antiwaterlogging should be performed. Discussion
In this study， the relationship between increment of trees and precipitation was analyzed using simulation equations， thereby opening up a new way. Due to unprofessional inaccuracy measurement and less collected data， many scholars concluded that tree growth is not closely related to precipitation. The results of this study showed that the selection of dimension did not affect test results， various linear and nearlinear equations had certain differences in test results and precision， but could interpret different problems， respectively， and the test precision had no big differences； and the longterm fitting equation had higher test precision than shortterm test precision， and could better illustrate the problem. Therefore， selecting longtime sample value is very important. This study adopted the data of analytic tree and used the empirical equations of tree growth， so the noise on test results caused by spatial and temporal differences of various natural conditions and tree differentiation was avoided. After all， the conclusions were obtained from the study on single tree， and should be verified in research and practice repeatedly， and they could only be put into use after the identification by relevant experts. However， this research would consume massive manpower and material resources， and requires at least several decades or even hundreds of years， and the research method in this study can yet be regarded as a simple rapid reliable method which could be popularized in future scientific research and production practice.
References
[1]CHENG HH. Mathematic statistics[M]. Beijing： China Forestry Publishing House， 1985.
[2]CHEN F， YUAN YJ， WEI WS， et al. Reconstruction of annual precipitation in Shandan based treering since 1783 A.D.[J]. Geography and GeoInformation Science， 2010（5）： 82-86. （in Chinese）
[3]CAO SJ， CAO FX， QI CJ， et al. Advances in research on the relationships between climatic change and pathological treering structures[J]. Ecology and Environmental Science， 2010， 2： 494-498. （in Chinese）
[4]CHEN JJ， SUN Y， HE XY， et al. Chinese pine treering width chronology and its relationships to climatic conditions in Qianshan Mountains[J]. Chinese Journal of Applied Ecology， 2007， 10， P2191-2201. （in Chinese）
[5]WANG YQ. Shandong Vegetation[M]. Jinan： Shandong Publishing House of Science and Technology， 2000， China. （in Chinese）
[6]ZHANG DQ， NIU JZ， CHEN ZM， et al. Study on spiral rise theory in ecological succession[J]. Protection Forest Science and Technology， 2002， 1： 14-16. （in Chinese） [7]WANG YQ. Comparative study on the vegetation between shandong and Liadong Peninsula[J]. Acta Phatoecologica at Geobotanica Sinica，1984，1： 41-50. （in Chinese）
[8]GU FT. The shell mounos in north Shandong Province and shell muund vegetation[J]. Acta Phatoecologica at Geobotanica Sinica，1990，3： 275-279. （in Chinese）
[9]JIAO QH，FENG JH. Discussion on recovery pattern of mountains forest ecological vegetation in Shandong Province[J]. Shandong Forestry Science and Technology， 1999， Supplement： 113-114. （in Chinese）
[10] ZHANG DQ， LI HR， MENG FZ. Discussion on recovery pattern of ecological vegetation in the Yellow River Delta[J]. System Sciences and Comprehensive Studies in Agriculture， 1999， Supplemen： 413-415. （in Chinese）
[11]ZHANG DQ， ZHANG YP，WANG SL， et al. Discussion on managing pattern of small valley in Shandong Province[J]. Forestry Science and Technology， 2000， 2： 23-24. （in Chinese）.
Editor： Yingzhi GUANG Proofreader： Xinxiu ZHU
Key words Threshold； Limit value； Empirical equation； Analytic tree； Annual precipitation
In various production practices of planting industry， waterlogging is a common problem， that is to say the uneven spatial and temporal distribution of precipitation would result in adverse effect on production， especially in forestry production. Many scholars concluded that the change of annual precipitation has no significant effect on tree growth according to growth ring information and annual precipitation information. The writers deem that the conclusion is not accurate， and might be due to inaccurate measurement （due to false annual ring）， or noncorrespondence between precipitation and local information， or the comprehensive effect of other noises. Also， some scholars concluded that precipitation is positively or negatively correlated with tree growth from local information， and some scholars reckoned the precipitation of the previous year has a more significant effect on tree growth in next year. In this study， the relationship between tree growth and annual precipitation was studied using the data of analytic tree taking Larix gmelinii as an example， so as to verify above viewpoints.
Data Source
The precipitation information over the years was provided by the meteorological department. The analytic tree information was collected from a 24yearold L.gmelinii growing normally in Laoshan forest farm in Laoshan district， Qingdao City on September 17， 1983. The breast height section was segmented according to 2.6 m， and other sections were segmented according to 2 m. Discs were cut at the tree heights of 5 cm （0 plate）， 1.3， 3.6， 5.6， 7.6， 9.6， 11.6， 13.6 and 14.6 m， respectively， and interpretation was performed strictly according to the technical points of analytic tree. Related information was collected with two years as an age class. Research Method
In order to save research cost， fitting tests were carried out to various regression equations using tree growth empirical equations according to "Forest Mensuration" and "Mathematical Statistics" edited by Northeast Forestry University， and with reference to the research results of Wang et al.[5-11]， and finally， the mixed type empirical equation， y（t）=ea-b/t （a and b are tobedetermined exponential parameters， and e is the base number of natural logarithm： 2.718 28……）， was selected for the study of tree growth process. Because tree growth is influenced by various factors， and the amount and spatial and temporal distribution of precipitation has the highest effect on L. gmelinii， the writers tried to fit the tree growth process using empirical equations， and through the acquisition of the accumulated annual precipitation according to DBH fitting equations （including the equation obtained in derivation， as described in the research process）， various precipitation indices affecting growth of treemeasuring factors including the diameter at breast height was obtained. The threshold value of annual precipitation for tree growth was calculated using y（p）=a-b/p， y（p）=a+bp and y（p）=a+blnp， and the threshold value of annual precipitation limiting tree growth was calculated using y（p）=1/（a-blnp） and y（p）=1/（a-bp）.
Research Process
Through the check analysis of tree growth equation fitting within two years， three treemeasuring factors （tree height， DBH and volume of timber） all had fitting relation with precipitation. Unary linear equations （direction application of unary linear equations） were formed by various arithmetical operations on various fitting equations. Then， parameters a and b were obtained by unary linear regression， and F test and correlation coefficient （R） test were carried out. If the tests are passed， a tree growth fitting equation is established， and the calculation results are shown in Table 1. Table lookup showed that the tests were passed with a reliability over 90%， indicating that these mathematical models （i.e.， empirical fitting equations） are applicative overall. The problem of solving the maximum current annual increment time and quantitative maturity of trees using fitting equations was illustrated taking the DBH item as an example. The precipitation fitting mixed equation of tree DBH growth y（p）=ea-b/p （the increment changing with precipitation， obtained by derivation of Y（t） function， the process of which was omitted， only the extreme point was given， the same below） showed the extreme value pz=272.6 mm， i.e.， when the accumulated precipitation reaches 272.6 mm in two years， the current water resourcecontributed increment reaches its peak （i.e.， the instant use efficiency of water is the highest）， which is the only peak； and the average water use efficiency equation of average tree growth Y（p）/p （precipitation converted from water resource consumed by annual average unit increment， the calculation process of which was omitted， only the extreme value was given， the same as below） had the extreme value pm=545.1 mm， i.e.， when the accumulated precipitation reached 545.1 mm in two years， the current average water resource use efficiency was the highest. Only the calculation of DBH fitting equation was described above， other fitting equations were the same as it， and detailed description was omitted. In the discussion below， growth fitting equations， and symbols such as pz and pm， also had the same meaning， and the calculation results were given without repeated description and explanation. The various calculated parameters and various threshold values are shown in Table 1. It could be seen from Table 1 that in the equation y（p）=ea-b/p， compared with DBH， various parameters of the square of DBH were improved by one time， their precision， F values and R values were equal， which was caused by the exponent mathematic relation， and in order to compared with accumulation， a fitting equation was also established with the square of DBH as a treemeasuring factor. Various equations all could pass F test and R test， and the precision is shown in Table with a lowest value of 90%， and the highest value of 99.9%. For ease of listing， symbols were used， for instance， 139.8 mm （±2 a） refers to that within two years， if the annual average precipitation is lower than 139.8 mm， positive growth turns to negative growth， and this value is called threshold value； 1 974.23 mm （±∞2 a） means that within two years， when the effective annual average precipitation reaches 1 974.23 mm， theoretical unlimited growth， the maximum increment is reached， but once the value is exceeded， immoderately destructive negative growth occurred， so the value is also known as threshold value （the same below）.
It could be seen from the test results that the change in precipitation had greater effect on the change of volume of timber before 16 a， the water demand was larger in early period， and water was the main limiting condition for tree growth； and in 13-15 a which was the peak period of tree growth， precipitation had greater effect on tree height； and in 6-24 a， precipitation had greater effect on the growth of DBH.
It could be seen from the test results that tree height exhibited negative correlation， and DBH and volume of timber showed periodic positive and negative correlation， and thus had no close relation to the change of annual precipitation.
It could be seen from Table 1 that equations V5 and V6 showed more proximate threshold values， with a relative error limit only of 1.82%， the test precision reached 98% and 99.9%， respectively， and therefore， these test results are more reliable. Equations V1 and V4 exhibited more proximate threshold values， with a relative error limit only of 0.44%. According to the fitted values with higher precision， the threshold value for tree growth was annual precipitation from 461.7 to 1 044.7 mm， exceeding which negative growth of trees， even daeath， might be caused. The equation of D21.3 also had the test precision of 99%， and D21.3 was in nearlinear correlation with the volume of timber. Through the derivation operation of the equation， the threshold value pz=545.1 mm was obtained， indicating that within two years， if the annual average accumulation reaches this value， the current growth rate of D21.3 is the highest； and the other threshold value pm=1 090.2 mm was also obtained， indicating that within two years， if the annual average accumulation reaches this value， the annual average growth rate is the highest. Obviously， the Pm value exceeded the threshold value 1 044.7 mm， but the two values were proximate， so the value with higher precision was chosen， the threshold value for tree growth was corrected to be from 461.7 to 1 090.2 mm. Through the check analysis on fitting of the growth process equations of the main treemeasuring factors within 24 years， all the treemeasuring factors all showed equation fitting relation， with very high reliability and correlation coefficient. The precision was about 99% mostly， nearly approximate to 100%. Various equation relation and values are shown in Table 2 （D1.3， H and V represents DBH， tree height and volume of timber， respectively， the same below）. The meanings of the values were the same as those of the DBH fitting equations within two years. Some of the precision in Table 2 is 100%， which means that the value is infinitely approximate to 100%， while the correlation coefficient is exactly 100% when the value is 1. However， due to the influence of subjective and objective noise， the fitting precision hardly could reach 100% （the same below）， and a higher fitting precision indicates the accuracy of the data and the applicability of the selected equation. Therefore， the test results in this study have higher scientific value. It could be seen from Table 2 that all the treemeasuring factors were in positive correlation with precipitation， and it was speculated that the periodic alternation of positive and negative correlation in individual years might be related to the unstable yielding phenomenon in seed propagation of trees. The growth and development of L. gmelinii is mainly vegetative growth， and therefore， the fitting equations of accumulated growth exhibited positive correlation in the whole process.
It could be seen from Table 2 that equations V1 and V2， and V3 and V4 showed more approximate threshold values， with relative error limits of 1.46% and 1.94%， respectively， and the test precision was approximate to 100%. Therefore， this group of test results were more reliable. The fitted values with higher precision were selected， and the threshold value of precipitation for tree growth was from 349.9 to 1 132.8 mm， beyond which negative growth of trees， even death， might be caused. The equation of D21.3 also had precision approximate to 100%， and D21.3 was in nearlinear correlation with the volume of timber. Through the derivation operation of the equation， one extreme point was obtained as pz=700.3 mm， which meant that when the average annual accumulation of precipitation reaches this value within 24 a， the current growth rate of D21.3 would be the highest； and the other extreme point was obtained as pm=1 400.6 mm， which meant that when the average annual accumulation of precipitation reaches this value within 24 a， the average annual growth rate would the highest. Obviously， the Pm value exceeded 1 132.8 mm， but the two values were close， so the value with higher precision was chosen， i.e.， the limit value of tree growth was corrected as from 349.9 to 1 132.8 mm. If irrigation is allowed， it is necessary to irrigate the trees when the annual precipitation is lower than 349.9 mm， mainly in the modes of sprinkling irrigation and drop irrigation. The optimal irrigation amount is equivalent to the annual precipitation of 700.3 mm （including natural precipitation， the same blow）， and the largest amount should not exceed 1 400.6 mm， beyond which the problem of drainage would be caused. The fitted vale of the equation H7 was extremely approximate to the pm value obtained from D1.3=exp（a-b/p）， and the two values were nearly the same， indicating very high test precision. The fitted values of equations V7 and V8 were far different from the actual conditions， but showed the trend that the water demand for tree growth increased with the age increasing. The calculation results of the materials within 24 a including the fitting results of the materials within 2 a exhibited improved test precision， and confirmed each other.
Through check analysis， when comparing current precipitation with the precipitation of the previous year， the F value was 1.333 （2.32 was the F value under the reliability of 90%）； and within 24 years， when comparing the accumulated value of d1.3 of the previous year with the fitted value of the equation d1.3=exp（a-b/p） of the current year and the actual value of another sample tree， the F values were 1.206 and 1.141， respectively， indicating there were no significant differences， and therefore， not only the fitting equation was established， but also the effects of current precipitation and the precipitation of the previous year on tree were hard to be differentiated. The data of the previous years were also subjected to fitting analysis， and except that the test precision of the previous year was slightly higher than the current year， no big differences were found. Therefore， repeated description was omitted. Some scholars deem that there is a lack of basis for the speculation that the precipitation of the previous year has a more significant effect on tree growth in the current year， and the fact might be that the precipitation of the previous year would affect the reproduction growth of trees， and then indirectly affects the vegetative growth in the current year. However， the writers deem that this is one of the reasons for the periodic change of the positive and negative correlation with annual precipitation， but the effect is very little， and the main reason is the periodic development law of trees. Other equations were also subjected to suitability test by the same method， and the fitted values of the equations all had no significant differences from the indices of another sample tree. However， there was a significant difference between the P. thunbergii×P. densiflora at another location and the P. thunbergii at the same location. Therefore， different tree species could not use the same equation， because tree species differ in growth and development laws.
Conclusions and Application
The results of this study showed that various treemeasuring factors had very remarkably correlation with the amount of precipitation and its distribution. For the growth of volume of timber， under the condition of uniform spatial and temporal precipitation distribution， when the average annual precipitation reached 700.3 mm， the annual increment reached the maximum value， and the instant water use efficiency was the highest； and when the average annual precipitation reached 1 400.6 mm， the average annual increment contributed by unit water resource reached its maximum value， and the average water use efficiency was the highest. Within 24 a， when the currently precipitation was lower than 349.9 mm， trees grew negatively and even died； and when the average annual precipitation reached 1 400.6 mm， the volume of timber had an infinite increment theoretically， but once the threshold value was exceeded， destructive negative growth of trees occurred， i.e.， death of trees due to waterlogging （the threshold value of DBH was 1 457.9mm， and that of tree height was 1 134.9 mm）. The fitted values of volume of timber and tree height confirmed each other， indicating that the test was very accurate. Within 2 a， when the average annual precipitation reached 545.1 mm， the instant use efficiency of water resource was the highest； and when that annual average precipitation reached 1 090.2 mm， the average annual use efficiency of water resource was the highest. According to above results， it is advised to irrigate the tree properly in the mode of drop irrigation when the forecasted annual precipitation is lower than 545.1 mm， and the largest amount of irrigation should not exceed 1 090.2 mm （including natural precipitation）. When the average annual precipitation exceeds 1 090.2 mm， drainage and antiwaterlogging should be performed. Discussion
In this study， the relationship between increment of trees and precipitation was analyzed using simulation equations， thereby opening up a new way. Due to unprofessional inaccuracy measurement and less collected data， many scholars concluded that tree growth is not closely related to precipitation. The results of this study showed that the selection of dimension did not affect test results， various linear and nearlinear equations had certain differences in test results and precision， but could interpret different problems， respectively， and the test precision had no big differences； and the longterm fitting equation had higher test precision than shortterm test precision， and could better illustrate the problem. Therefore， selecting longtime sample value is very important. This study adopted the data of analytic tree and used the empirical equations of tree growth， so the noise on test results caused by spatial and temporal differences of various natural conditions and tree differentiation was avoided. After all， the conclusions were obtained from the study on single tree， and should be verified in research and practice repeatedly， and they could only be put into use after the identification by relevant experts. However， this research would consume massive manpower and material resources， and requires at least several decades or even hundreds of years， and the research method in this study can yet be regarded as a simple rapid reliable method which could be popularized in future scientific research and production practice.
References
[1]CHENG HH. Mathematic statistics[M]. Beijing： China Forestry Publishing House， 1985.
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Editor： Yingzhi GUANG Proofreader： Xinxiu ZHU