全 文 ：Growth Analysis of Individual Tree Basal Area of Western
Yellow Pine Introduced in Kostelec Region

LIAO Chao-ying
1
,　 PODRA′ ZSKY′ Vile′m2
( 1. Col lege of Resources an d Environment , N W Sci-Tech Univ. of Agr . and For . , Yang ling , Shaanxi 712100, China;
2. Facu lty of Forest ry , Czech Universi ty of A griculture, Kamy′cká , Prague 165 21, Czech )
Abstract: Four theo retical g row th functions, the Mitscherlich, Logistic, Gompertz and Ko rf functions
w ere applied to the basa l area g row th data of individual Western yellow pine ( Pinus ponderosa ) t rees to
model the basal area g row th. The current increment , mean increment and grow th intensi ty of basa l a rea as
functions o f age were giv en by deriv ation f rom the best fi t g row th function. The g row th process of basal
a rea was divided into three periods by the tw o inflexion points of the current increment curv e of basal area.
The results indicated tha t the Ko rf function fi t ted the basal a rea g row th best , fol low ed by the Gompertz,
Logistic and Mitscherlich functions. The maximum values of the current increment and the mean increment
appeared a t ag e 7 years and age 15 yea rs respectiv ely. The average g row th intensi ty o f basal area w as
0. 243. During the rapid g row th period ( 3 to 11 yea rs) the average increment of basal a rea per y ear w as
0. 001 474 m
2
.
Key words: W estern yellow pine (Pinus ponderosa Dougl. ) ; basal area; grow th analysis; Gompertz func-
tion; Ko rf function; Logistic function; Mi tscherlich function
CLC number: S758. 1　　 Document code: A　　 Article ID: 1001-7461( 2001) 04-01-05
KOSTELEC地区西黄松林木个体胸高断面积生长分析
廖超英 1 ,　 PODRA′ ZSKY′ Vile′m2
( 1. 西北农林科技大学 资源环境学院 ,陕西 杨陵 712100; 2. 捷克农业大学 林学院 ,捷克布拉格 165 21)
摘　要: 用 Mitscherlich、 Logistic、 Gompertz和 Korf 4个理论生长方程分别对引种到捷克 Kostelec地区的
西黄松林木个体胸高断面积生长过程进行了拟合 ,由最佳生长方程分别得出了胸高断面连年生长量、平均生
长量和相对生长率随时间变化的函数 ,并对生长过程进行了分析。 结果表明: Ko rf方程具有最高的拟合精
度 ,能很好地描述西黄松胸高断面积生长过程 ;西黄松胸高断面积连年生长量和平均生长量最大值分别出现
在第 7 a和第 15 a(胸高年龄 ) ; 1～ 26 a的平均相对生长率为 0. 243;速生期 ( 3～ 11 a)内胸高断面积平均生
长量为 0. 001 474 m2。
关键词:西黄松 ; 胸高断面积 ; 生长分析 ; Gompertz方程 ; Ko rf方程 ; Logistic方程 ; Mitscherlich方程
　　 The western yel low pine ( Pinus ponderosa
Doug l. ) i s na tiv e to th e Rocky Mountains of No rth
America, ranging f rom Mexico in the south ( 23°
N ) through Cali fo rnia and Oregon up to Briti sh
Columbia in Canada ( 52°N ) . It is found a t elev a-
tions of 300 to 2 100 m pa rticularly in drier regions
( annual precipi ta tion 400～ 700 mm). In i ts nativ e
land trees 40～ 50 m tall wi th diameters of 3 m
have been reco rded. The deep roo t sy stem enables
i t to endure long-lasting drought and it can g row
西北林学院学报　 2001, 16( 4): 1～ 5　　
Journal o f Nor thw est Fo restr y Univ ersity
　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　
Received date: 2001-05-06
Foundation item: Th e res earch project ( MSM 414100009) su bsidized by the Minis t ry of Envi ronmen t of the Czech Republic
Biography: Liao Ch ao-ying ( 1959-) , male, native of Anhui. Ph. D. , as sociate profess or, Research f ield: si lviculture and fores t ecology.
on rela tiv ely poo r, dry soi ls. The species w as in-
troduced to Europe in 1827, and today is widely
planted in forest plantations and in parks. In this
paper the Mitscherlich, Logistic, Gompertz and
Ko rf functions w ere used to model the basal a rea
g row th of stand average t rees cho sen from 4 even-
aged w estern yellow pine stands in Kostelec o f the
Czech Republic, the best fi t g row th function was
selected, and the g row th process o f basal area wa s
analy zed.
1　 Natural Conditions
The w estern yellow pine stands are lo cated
near the tow n Kostelec nad C ern y′mi lesy in the
Czech Republic. Geog raphical po si tio n is: longi-
tude 14°51′E and lati tude 50°01′N. M ean annual
tempera ture, mean temperature of January and
mean temperature of July a re 8. 1℃ , - 1. 9℃ and
17. 8℃ respectiv ely. The highest temperature is
40. 8℃ and the low est is - 28. 5℃ . M ean annual
precipi tation is 662. 6 mm. The geological bo t tom
is Permian and chalk sand stone covered wi th thick
lay er of loam loess. Acco rding to silvicul tural clas-
si fica tion, the area is classified as acid beech oak
fo rest ( Roc ek et al. , 1998) .
2　Material and Meth ods
2. 1　Material
Four research plo ts w ere established in 4
even-aged pure w estern yellow pine stands, the ar-
eas of the plo ts ranging from 0. 08 to 0. 11 hm
2 .
Diameter a t breast height (dbh ) and tree height as
w ell as crow n class w as recorded for each tree in
the plo ts. In each plot one t ree w hose dbh was
most clo se to the quadratic mean diameter o f the
plo t w as chosen as sample t ree ( stand average
t ree) , the sample t ree was felled and a complete
stem ana lysis ( Avery et al. , 1983) w as made.
The measurement of annual ring s w as made to
a precision of 0. 01 mm with CODIM A ( an incre-
ment measuring dev ice equipped wi th a microsco-
pe) .
Prog rams EXCEL, S TATGRAPHICS and
SAS were used for data processing.
2. 2　Methods
2. 2. 1　 Fit of Grow th Functions
The Mitscherlich, Logistic, Gompertz and
Korf g row th functions w ere fi tted to the basal area
( ba) g row th da ta:
Mi tscherlich　 y= A ( 1- e- bt ) , ( 1)
Logistic y=
A
1+ e
a - bt , ( 2)
Gompertz y= A e
- ea- bt , ( 3)
Korf y= A e
b
( 1- a )ta- 1 , ( 4)
w here y i s the ba at age t , e i s the base of natural
log ari thms, a and b are the parameters to be deter-
mined by the method o f “ least squa res” fit ting fo r
the g row th da ta, A i s asympto tic ba—— the point
w here basal area g row th equals zero and i t w as es-
tima ted by the“ three points” method.
The Mitscherlich curv e is characterized by no t
having any inflexion w ith asymptote at f ( t )= A,
w hile the Logistic, Gompertz and Ko rf curv es are
sigmoid curv es wi th asympto tes a t f ( t ) = 0 and
f ( t ) = A. The Logistic and Gompertz curv es are
characterized by an inf lexion appearing a t ex actly
the halfw ay and approximately one-thi rd points of
the entire g row th pro cess respectiv ely ( Sw eda et
al. , 1981) .
The goodness of fi t w as evalua ted by R2 o r
M SSD ( Sw eda et al. , 1984):
R
2
= 1 -
∑n
i= 1
( Yi - yi )
2
∑n
i= 1
(Yi - Y- )
2
, ( 5)
MSSD = 1
n - f∑
n
i= 1
( Yi - yi ) 2 , ( 6)
w here R
2: co ef ficient of determination,
MSSD: mean squared sum of deviations,
Yi: o bserv ed ba at ag e i ,
yi: calculated ba a t ag e i ,
Y-: average of the actua l observa tion,
n: total ag e of th e sample t ree,
f : number of parameters involv ed in the equa-
tion concerned, tha t is, tw o fo r the Mitscherlich
and three fo r the o thers.
The denomina to r n-f in the above expression
ensures a fair comparison among g row th equations
2　　　　　　　　　　　　　　　　　　　　　　西北林学院学报　　　　　　　　　　　　　　　　　　　　　 16卷
w ith dif ferent number o f parameters.
2. 2. 2　 Analysis o f Grow th Process
2. 2. 2. 1　 Current Annual Increment and Grow th
Intensity
Di fferentia ting the best fi t g row th function, y
= f ( t ) , wi th respect to age giv es the current annu-
al increment G and dividing G by the existing y
giv es g row th intensity R ( the ratio o f increment of
ba to ba i tself ) , that i s,
G=
dy
dt
= f′( t ) , ( 7)
R = G
y
= f′( t )
f ( t )
. ( 8)
　　 If the second deriv ativ e o f the g row th function
a t the age t1 is 0, the cur rent annual increment G
gets i t s maximum value. i. e. when f″( t1 )= 0, G=
f′( t 1 )= max.
The mean g row th intensi ty R- ( Causton, 1981)
during the interv al tj- ti is
R- =
lnYj - lnYi
tj - ti . ( 9)
　　 For Korf g row th function,
G=
Ab
ta
eb( 1- a) - 1t1- a , ( 10)
i ts maximum va lue　Gmax= Aae ·
a - 1
a
eb
, ( 11)
t1 =
　
a- 1 b
a
, ( 12)
R =
b
ta
. ( 13)
2. 2. 2. 2　 Mean Annual Increment (MAI ) and Its
Culmination Age
MAI=
y
t
= f ( t )
t
. ( 14)
Maximum of the mean annual increment is at
the age t 2 , w hen the fi rst deriv ativ e of the function
o f mean annual increment is 0, i. e. (
f ( t2 )
t2
)′= 0.
Fo r Ko rf function,
MAI =
A
t
eb ( 1- a )
- 1t 1- a , ( 15)
t 2 =
a - 1
b . ( 16)
2. 2. 2. 3　 Division of the Grow th Process
Inf lexion points of the current annua l incre-
ment curv e are a t the ages t31 and t32 , at w hich the
thi rd deriv ativ e of the g row th function equals zero
( i. e. y″′= 0) and the g row th ra tes change most
rapidly. The g row th process is divided into three
periods ( Liao et al. , 2000): [0, t31 ) , [t31 , t32 ) a nd
[t 32 , + ∞ ) .
Fo r Ko rf function,
t31 , t32 =
　
a- 1
b 3 5 -
4
a
2(a + 1)
. ( 17)
　　 In the f irst period, [0, t31 ) , the g row th rate is
low er and the g row th rate curve is concave. This
period can be called “ pre-rapid” grow th period.
The second period, [t31 , t32 ) , wi th the highest
g row th ra te can be called rapid g row th period, in
w hich the g row th ra te curv e appea rs convex. Dur-
ing the thi rd period, [t 32 , + ∞ ) , the g row th rate
gets low er and the g row th ra te curve is concave, so
the period can be cal led “ po st-rapid” g row th peri-
od.
3　 Results and Discussion
3. 1　 Growth Functions
The basal a rea g row th da ta of the sample t rees
w ere fi tted to the Mitscherlich , Logistic, Gom-
pertz and Korf functions, and the goodness of fi t
w as examined by R
2
and MSSD. The pa rameters
fo r each g row th function and the indicato rs of fi t
are giv en in table 1, and the curv es o f these g row th
functions a re show n in figures 1 through 4.
Table 1　 Parameters of the growth functions and the indicators of f it
Type a b A Grow th function R 2 MSSD
Mitscherlich 0. 074 3 0. 029 103 y= 0. 029 103( 1- e- 0. 074 3t ) 0. 866 4 1. 28E- 5
Logistic 3. 954 2 0. 291 5 0. 027 16 y=
0. 027 16
1+ e3. 954 2- 0. 291 5t
0. 951 9 4. 2E- 6
Gom pert z 1. 620 9 0. 152 1 0. 029 103 y= 0. 029 103e- e
1. 620 9- 0. 152 1t
0. 994 1 5. 18E- 7
Korf 1. 827 4 8. 985 4 0. 055 610 y= 0. 055 610e
- 8. 985 4
0. 827 4t
0. 827 4 0. 999 7 2. 45E- 8
3第 4期　　　　　　　　　　廖超英等　 KOSTELEC地区西黄松林木个体胸高断面积生长分析　　　　　　　　　　　　
Fig. 1　 The Mitscherlich curve ( solid line) compared
with the observed ba growth ( white dots)
Fig. 2　 The Logistic curve ( solid line) compared
with the observed ba growth (white dots)
Fig. 3　 The Gompertz curve ( solid line) compared
with the observed ba growth ( white dots)
Fig. 4　 The Korf curve ( solid line) compared
with the observed ba growth (white dots)
　　 As seen in table 1, the Ko rf g row th function
achiev es the best fi t w ith the larg est R2 and the
smallest MSSD , follow ed by the Gompertz and
then by the Logistic, and the Mitscherlich function
fi t s the basal a rea g row th w ith the least accuracy.
3. 2　 Current Annual Increment, Mean Annual In-
crement and Growth Intensity
The first deriv ativ e of the best fi t grow th
function was used to express the current annual in-
crement o f basal area. Substi tuting the parameters
in equations ( 10) , ( 11) and ( 12) wi th the cor re-
sponding values f rom table 1, the current annual
increment o f basal a rea as a function o f age would
be
G=
0. 499 678
t 1. 827 4
· e- 10. 860 4t- 0.827 4 , ( 18)
and its maximum value (Gmax ) and the culmina tion
age ( t1 ) would be
Gmax =
Aa
e

　
a- 1 a
eb
= 0. 001 628 ( m2 ) ,
t 1 =
　
a - 1 b
a
= 6. 86 ( a) .
In o ther w ords, the current annual increment of
basal area w as maximum ( 0. 001 628 m
2 ) at the
age o f 7 yea rs ( breast height ag e) .
According to equa tions ( 15 ) and ( 16) , the
mean annual increment of basal area as a function
of ag e is giv en by
MAI =
0. 055 610
t e
- 10. 860 4t- 0. 827 4
, ( 19)
and the culmina tion age of mean annual increment
of the basal area w ould be
t2 =
a- 1
b = 14. 21 ( a) .
　　 The curv es o f equation ( 18) and equation ( 19)
are show n in figure 5. As can be seen in the fig-
ure, the tw o curv es crossed at the age 15, and at
the same time the mean annual increment of basal
area go t the maximum value ( 0. 001 169 m
2 ) .
4　　　　　　　　　　　　　　　　　　　　　　西北林学院学报　　　　　　　　　　　　　　　　　　　　　 16卷
Fig. 5　 The curves of current annual increment
and mean annual increment of ba
Fig. 6　 The growth intensity of ba as a funct ion of age
　　 Grow th intensity (R ) is the ratio of the fi rst
deriv ative of the g row th function to the grow th
function i tself , o r the cur rent annual increment per
uni t basal area. Acco rding to equa tion ( 13 ) ,
g row th intensi ty of basa l a rea as a function o f age
is
R =
8. 985 4
t
1. 827 4 . ( 20)
　　 The curv e o f the function ( 20) is show n in fig-
ure 6. It can be seen tha t the g row th intensity de-
creased rapidly at early stage, and then the de-
crease go t slow er and slow er.
The mean g row th intensi ty of basal area dur-
ing age 1 to 26 w as
R- = ln0. 026 619 - ln0. 000 061
26 - 1
= 0. 243.
3. 3　 Division of the Growth Process of Basal Area
Substi tuting a and b in equa tion ( 17) w ith the
cor responding values f rom table 1, the ages t31 and
t32 , at which the inf lexion points of the current an-
nual increment curv e appeared, would be
t31 = 2. 46 ( a) ,
t32 = 11. 29 ( a) .
Therefore, the basal area g row th pro cess w as di-
vided into the following three periods: 0 to 2
yea rs, 3 to 11 yea rs and 12 years up. During the
rapid g row th period ( 3 to 11 years) , the average
increment of basal area per yea r w as
f ( t32 ) - f ( t3 1 )
t 32 - t31
= 0. 001 474 m2 .
4　 Conclusions
The application of the Mitscherlich, Logistic,
Gompertz and Ko rf functions to the observ ed basal
area g row th data o f individual w estern yellow pine
t rees rev ealed tha t the Ko rf function fi tted the
basal area g row th best and expressed the basal
area g row th process remarkably w ell , follow ed by
the Gomper tz function and then by the Logistic
function, and th e Mitscherlich function fit ted the
basal area g row th wi th the least accuracy.
The current annual increment of basal area as
a function o f ag e w as G =
0. 499 678
t
1. 827 4 ·
e
- 10. 860 4t- 0. 8274 , it s max imum value ( 0. 001 628 m
2 )
appea red a t the age 7 yea rs. The mean annual in-
crement of basal area as a function o f age w as MAI
=
0. 055 610
t
· e- 10. 860 4t- 0. 827 4 , and it go t the maxi-
mum value ( 0. 001 169 m
2 ) at the age o f 15 years.
Grow th intensi ty as a function o f age was R =
8. 985 4
t
1. 827 4 , a nd the average g row th intensi ty o f basal
area during age 1 to 26 w as 0. 243.
There are dif ferent w ays to divide g row th pro-
cesses. In this paper, the two inflexion points of
the current annual increment curv e w ere used to
divide g row th pro cess into three periods: 0 to 2
years, 3 to 11 years and 12 yea rs up. During the
rapid g row th period ( 3 to 11 years) the average in-
crement of basal a rea per yea r w as 0. 001 474 m
2
.
Acknowledgments:
The au thors gratefully ack nowledg e M r. Kratochvil and Engi-
neer Ji r

i Rems

e for thei r h elp in fieldw ork.
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5第 4期　　　　　　　　　　廖超英等　 KOSTELEC地区西黄松林木个体胸高断面积生长分析　　　　　　　　　　　　
同时观察生长状态发现 ,干旱对照虽然有一定的存
活率 ( 30%～ 40% ) ,但大部分叶子干枯 ,而固体水处
理的植株生长状况基本正常。
表 5　固体水对树木幼苗成活率的影响
Table 5　 Ef fect of s olid water on su rviving
rate of th e plants %
树种 处 理水 平
I II III IV V
海南蒲桃 40 80 100 100 100
红花油茶 45 80 100 100 100
土 沉香 30 70 95 100 100
阴香 20 65 95 100 100
3　结论与讨论
4种供试树苗经春之霖固体水处理后 ,其土壤
含水量、叶含水量、叶绿素含量、高生长量和植株成
活率都明显高于干旱对照 ,且各指标的增量与所使
用的固体水截面积呈正相关。这表明固体水在干旱
条件下对于维持土壤有效水分 ,提高树木幼苗的成
活率以及维持植株的正常水分状况和代谢水平具有
显著的效果 ,因此在荒山造林和荒漠绿化生产实践
中具有重要的意义和推广价值 ,是干旱地区造林的
最佳供水途径。
在实际生产中 ,固体水的用量需要考虑植物本
身的水分生理生态特性和植株的生长状况 (叶量的
多少 )以及土壤的类型。对于喜湿、生长状况好 (总叶
面积大 )的苗木需要的固体水截面积要大一些 ;而相
对耐旱、叶面积小的苗木可适当减少固体水用量。
不同类型的土壤 ,其有效水分的范围 (永久萎蔫
系数 )不同 ,要使植物能够存活和正常生长 ,土壤含
水量应在有效水范围内。 对于本文所用的中壤土而
言 ,在连续 30 d干旱条件下 ,处理 II(固体水截面积
为 20 cm2 )的土壤含水量已接近永久萎蔫系数 ,因
此 ,其使用量应不低于截面积 20 cm2的固体水。
参考文献:
[ 1 ]　朱广廉 ,钟文海 . 植物生理学实验 [M ]. 北京: 北京大学出版
社 , 1990. 67～ 70.
[2 ]　劳家柽主编 , 土壤农化分析手册 [M ]. 北京: 农业出版社 ,
1988. 209～ 210.
[ 3]　中国科学院上海植物生理研究所 . 现代植物生理学实验指南
[M ] . 北京: 科学出版社 , 1999. 95～ 96.
(上接第 5页 )
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[1 ]　 Av ery T E, Burkh art H E. Fo res t measu rements [M ]. 3rd
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[2 ]　 Caus ton D R. The biomet ry of plant grow th [M ]. London:
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12　　　　　　　　　　　　　　　　　　　　　　西北林学院学报　　　　　　　　　　　　　　　　　　　　　 16卷