THE CONSTRUCTION OF TAPER CURVES 1 
By Frederick S. Baker 
Forest Examiner , Forest Service, United States Department of Agriculture 
INTRODUCTION 
Among the first requisites of a 
forester’s equipment in estimating 
standing timber are accurate and de¬ 
pendable volume tables, applicable to 
the stands being estimated and the 
utilization expected. Such tables show, 
in general, the average volume (in 
board feet, cubic feet, or other units 
of measure) of trees of different diame¬ 
ters and heights. It has not proved 
easy to construct dependable tables of 
this kind, owing to the difficulty of 
determining accurately the average 
volume of trees in the unusual size 
classes represented by very tall, slender 
trees and very short, large trees, for it is 
difficult to discover specimens enough 
of these classes to furnish the basis of 
reliable averages. 
The first volume tables to be used 
were constructed simply by scaling a 
large number of trees of different sizes, 
averaging the scale of all trees in the 
same diameter and height class and 
then harmonizing the values for the 
different height classes as well as possi¬ 
ble by graphic methods. This was 
the method described in 1906 by 
Graves (9, p. 158-163 , 166 , 167)? It 
is slow, requires a large mass of data, 
and the graphic harmonization neces¬ 
sary to make values run smoothly 
often introduces grave inaccuracy. 
In 1915, Barrows ( 1 , 2) described a 
new method, which introduced taper 
curves, or curves showing the average 
form of trees, as a step in volume table 
construction. This was a great im¬ 
provement over the earlier systems, 
and has been very generally used since 
that time. The method was accepted 
as undoubtedly sound and effective, 
the results were believed to be entirely 
satisfactory, and much time and labor 
were put in on the tedious series of 
curves necessary in this method. 
The present writer, however, in making 
a series of taper curves for lodgepole 
pine on a slender basis of trees, dis¬ 
covered that very palpable errors had 
come through the whole series of har¬ 
monizations and that the final curves 
were not dependable. This has led 
to an analysis of the underlying theory 
of Barrows’s series of recurvings, and 
to the conclusion that the latter are not 
as satisfactory as they would appear 
to be on the surface. It is no new dis¬ 
covery that errors exist in this method 
of handling taper curves. Barrows 
himself admits the fundamental error, 
that taper curves give not the average 
volume of the trees in any group, 
but the volume of the tree of average 
dimensions. 
In finding average tree form, diam¬ 
eters are averaged; in the earlier 
method, volumes proportional to diam¬ 
eters squared were averaged. The 
error introduced depends upon the 
range of values in a given class, but, 
as pointed out by Barrows, it never 
becomes serious. Bruce (4) implies 
more grave dangers by demonstrating 
the presence of errors in certain volume 
tables undoubtedly built up according 
to Barrows’s method. He shows that 
in these tables the values vary errat¬ 
ically from the volumes of the frustums 
of cones having the same top diam¬ 
eters and the same breast-high diam¬ 
eters as the trees; a divergence that 
can not possibly be true if the values 
pretend to represent true means. 
The cause of this variation, however, 
was never determined by Bruce, which 
leaves an uncertainty as to just what is 
wrong with taper curves, and whether 
the difficulty is remediable or not. 
NATURE OF UNHARMONIZED TAPER 
CURVES 
In analyzing Barrows’s method and 
the errors inherent in it, consideration 
must first be given to the basic data 
from which the taper curves are drawn. 
These consist of individual tree records 
giving diameter, usually by regular 
intervals (often 8 or 16 feet) from the 
stump to the top of the tree. Height 
is given to the nearest tenth of a foot; 
diameter to the nearest tenth of an 
inch. The size classes which are used 
as the basis of all computations, and 
which appear in the final volume tables 
1 Received for publication June 30, 1924; issued June, 1925. 
2 Reference is made by number (italic) to “Literature cited,” p. 624. 
Journal of Agricultural Research 
Washington, D. C. 
( 609 ) 
Vol. XXX, No. 7 
Apr. 1, 1925 
Key No. F-13 
