Apr. 1, 1925 
The Construction of Taper Curves 
621 
Table I. — Comparison of taper for Behre’s western yellow pine form class with the 
taper for Douglas fir , form quotient .700 
Per 
cent- 
age of 
length 
from 
tip to 
breast 
height 
Percentage of d. 
b. h. (inside bark) 
Per¬ 
cent¬ 
age of 
length 
from 
tip to 
breast 
height 
Percentage of d. 
b.h.(insidebark) 
West¬ 
ern 
yellow 
pine ° 
Douglas 
fir b 
West¬ 
ern 
yellow 
pine « 
Douglas 
fir b 
90 
95.5 
96.0 
40 
60.9 
61.2 
80 
90.3 
90.6 
30 
50.1 
50.5 
70 
84.5 
85.0 
20 
36.9 
37.8 
60 
77.7 
77.5 
10 
20.6 
21.5 I 
50 
70.0 
70.0 
« Behre. b Baker. 
TAPER CURVES BASED ON THE 
PARABOLOID FORMULA 
The system of curve construction 
just described was evolved because a 
still simpler system failed to work on 
Douglas fir, although it gave excellent 
results with both lodgepole pine and 
aspen. This involves the assumption 
that the taper curve follows the 
generalized formula for a paraboloid: 
Y — px n y or as it is more often written 
in forestry literature, Y 2 = px r . In 
this equation Y— diameter, x = dis¬ 
tance from the top of the tree, and p 
and r are constants. This formula 
gives a series of parabolic curves 
varying in shape with the value of r, 
and in steepness with the values of p. 
The first question that naturally 
arises is, how much risk of introducing 
■error we take by the use of this hy¬ 
pothesis. Judging from Graves's “ Men- 
with the actual average form in the 
case of the 60-foot height class in 
lodgepole pine. The curve Y 2 =px r 
was based upon d. b. h. and d. h/2 
in this work. Column A gives the 
actual average values (unharmonized 
by curves) in each class, column B the 
theoretical values. It is obvious that 
neither is an absolute value; there is a 
probable error in each. This has 
been figured for the actual values and 
it runs very close to ±0.1 inch, except 
in the 14-inch class, where it is nearer 
±0.2 inch. The probable errors of the 
computed values in columns B are of 
approximately equal magnitude. 
It is very evident that the error in 
assuming the trees to be paraboloids 
lies well within the probable error of 
±0.1. Hence it is obvious that the form 
of lodgepole pine is virtually a parab¬ 
oloid, at least in the 60-foot height 
class. Tests similar to that shown in 
Table II. — Comparison of theoretical form with the actual average form for 60- 
foot height class lodgepole pine 
Diameters at specified heights above stump (inches) 
D.b.h. 
(ins.) 
8 
16 
24 
32 
40 
48 
A 
B 
A 
B 
A 
B 
A 
B 
A 
B 
A 
B 
10 
9.6 
9.5 
8.9 
8.8 
8.2 
8.0 
7.2 
7.0 
5.9 
5.9 
4.3 
4.5 
11 
10.5 
10.5 
9.6 
9.7 
8.9 
8.8 
7.7 
7.8 
6.5 
6.5 
4.4 
5.0 
12 
11.4 
11.4 
10.5 
10.4 
9.3 
9.3 
8.1 
8.1 
6.6 
6.7 
5.1 
5.0 
13 
12.2 
12.3 
11.2 
11.1 
9.9 
9.9 
8.5 
8.5 
6.9 
6.9 
4.6 
5.0 
14 
13.0 
13.2 
12.2 
11.7 
10.4 
10.3 
8.8 
8.6 
6.9 
6.8 
4.8 
4.8 
A=Actual average values, unharmonized by curves. B=Theoretical values. 
•suration,” (9) it is used widely in all 
Europe, and exists in an implied or 
approximate form in many formulas 
for determining the volume of trees. 
Table II is presented to show the 
comparison of the theoretical form 
the above table were made in a variety 
of diameter and height classes where a 
large number of tree measurements 
were available and the results were 
equally conclusive in every case. 
Accordingly, it was assumed that the 
