《Table 2 Empirical equations for tree height growth process》

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《Study on Anticipant Mature Age of Pinus tabuliformis in Shandong Province》

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A linear equation was formed by taking the logarithm of the tree growth process equation.Then，parameters a and b were obtained by unary linear regression，and F test and significance test of regression coefficient were carried out.If t test is passed，a tree growth fitting equation is established，and the calculation results are shown in Table 1-Table 3.It might be due to the reason that the samples were not enough，the fitting precision of P.tabuliformis was greatly reduced compared with that of P.densiflora equation as control.The tree height equation had a very low precision in 2002，and was of no practical significance.The P.tabuliformis volume fitting equation had precision values lower than 90%in1997 and 2002，but passed F test and t test with the reliability over90%，indicating that the mathematical model（empirical fitting equation）is applicable overall.If the correlation coefficient（R）test is passed，then the fitting equation relation is established.The fitting degree of P.tabuliformis equation was lower in 2002 and1977，which might be due to that the number of samples was small，and therefore，fitting equations were established using all samples in 1988-1997 and 1988-2002，respectively，and compared with the study on P.densiflora in the same time period.For the 1988-1997 and 1988-2002 fitting equations，except that the tree height fitting equation of 1988-2002 had a precision reaching96%，the precision of other items all exceeded 99.9%，and the1988-1977 fitting equations had precision higher than the 1988-2002 equations.The problem of solving the maximum current annual increment time and quantitative maturity of trees using fitting equations was illustrated taking the DBH item of P.tabuliformis in1988 as an example.The tree growth rate equation of P.densiflora DBH（the current annual increment，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 tz=17.4 a，i.e.，the current increment of trees reached its peak value at about 18 a，and had only one peak value.The average growth rate equation Y（t）/t（annual average growth，the calculation process was omitted，only the extreme point was given，the same below）had the extreme value tm=34.7 a，i.e.，the quantitative mature age of trees was 35 a.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，the growth fitting equation，tree growth rate equation and tree average growth rate equation had the same meaning，symbols such as tz and tm also had the same meaning，and the calculation results were given without repeated description and explanation.The research results of P.densiflora were used as control of P.tabuliformis.The quantitative mature ages calculated according to various items are shown in tables.It could be seen from above tables that P.tabuliformis DBH and volume items had the highest precision in 1988，and the tree height item had the highest precision in 1997.The DBH fitting equations of 1992 and 1997 were more proximate，exhibiting a relative error only of 1.7%.Compared with P.densiflora，the fitting results of DBH equation in1988-2002 were more proximate to those of the P.densiflora equation in 1977，with a relative error only of 0.1%.The fitting precision of the volume of P.tabuliformis in 1988-1997 was surprisingly similar to that of P.densiflora in 2002，with a relative error only of 1.7%，namely P.tabuliformis and P.densiflora had the same mature age with a reliability of 98.3%.The fitting results of tree mature age，DBH and tree height equations were quite different from national standards，while for the fitting results of volume，P.tabuliformis was quite close to P.densiflora in 1997，and the mature age was slightly higher than national standard.Therefore，according to the research results and production practice，it is more reliable to use accumulative mature age as tree mature age.The fitting results of the volume equation of P.tabuliformis in 1988 showed a P.tabuliformis mature age of 74.9 a，with a fitting precision over 99.9%and correlation coefficient up to 0.96，which were far higher than national mature age standards，and the forest could be determined as the mature forest at ideal state，for special-purpose forest and protection forest（public welfare forest）.Furthermore，the fitting results of the volume equation of1988-1997 with the second best precision showed a P.tabuliformis mature age of 56.08 a，and the forest was determined as the mature forest at normal state，and used as the standard of mature forest for timber（commercial forest）.The results substantially accorded with the results of P.densiflora.The anticipant mature ages of public welfare forest and commercial P.densiflora forest were determined as 75 and 56 a，respectively.