Apr. 1, 1925 
The Construction of Taper Curves 
615 
smoothly from class to class. The 
system, however, can not claim a great 
degree of accuracy. In fact, it tends 
to sacrifice accuracy to harmony. To 
work best, not only- a large basis of 
individual measurements is needed, 
but they must be well distributed 
through many diameter and height 
classes so that the resulting curves may 
be long and their trend be clearly 
shown, minimizing the ever present 
danger of tilting the curves into in¬ 
correct positions. 
The object of this criticism of the 
method of preparing taper curves is 
not to throw doubt upon the accuracy 
of the volume tables prepared in ac¬ 
cordance with this system (which 
have proven fairly satisfactory) but to 
show that it is not by any means per¬ 
fect and that equal accuracy may be 
obtained by simpler means. 
FRUSTUM FORM FACTORS— 
bruce’s METHOD 
One obvious way of getting around 
the difficulties of Barrows’s method is 
to set a standard form of tree and com¬ 
pare all others with it. This is the 
essence of Bruce’s frustum form factor 
idea (4). 
The difference between the volume 
of a tree and the volume of a frustum 
of a cone having the same top diameter 
and the same d. b. h. ought to vary 
slowly and consistently with the 
changes of diameter, height, and form. 
Sudden changes are inconceivable 
where average trees are concerned. 
The frustum form factor method of 
volume table construction has been em¬ 
ployed by Bruce and has proven most 
excellent indeed. It is easy, quick, 
and, as demonstrated by Bruce and 
others, is surprisingly accurate (5, 7, 
10). It has the disadvantage, how¬ 
ever, of what may be termed inflexi¬ 
bility. The final results come in one 
step from the basic data, and if there 
are any other results desired they must 
be worked up anew from the original 
measurements. 
If a volume table is desired running 
to a 6-inch top limit, the volume of 
each tree to that limit must be figured 
and compared with the corresponding 
frustum. The frustum form factors 
must be averaged, and the computa¬ 
tion for volume be made from them. 
If now a new table is desired showing 
volume to a 7-inch limit, the whole 
process must be repeated. With taper 
curves, the preparation of new tables 
is a matter of minutes. Furthermore, 
the taper curve system is the only one 
applicable to linear products, ties, 
props, etc. 
Bruce’s frustum form factor is an 
empirical sort of figure bearing little 
relation to other usually accepted tree 
form constants, as form factor, form 
quotient, form exponent. One reason 
is that diameter breast high outside of 
bark is made one of the points through 
which the frustum surface passes, so 
that the frustum form factor will vary 
with bark thickness, if all other factors 
as height, diameter, and form remain 
constant. Thus it is not a measure of 
form in the sense that some other fac¬ 
tors are. 
The frustum form factor values also 
will vary with the top cutting limit 
used, because the lower part of a tree 
has more conical taper and the frustum 
of a tree cut to a 10-inch limit will fit 
much more closely to the frustum of a 
cone than when the top limit is 6 
inches, well in the top of the tree. It 
is possible for two tree frustums of dif¬ 
ferent form and equal top and basal 
diameters to have the same volumes, 
and hence the same frustum form fac¬ 
tors. This factor is thus an expression 
of volume relations rather than form 
relations. It is a very useful empirical 
figure, and the basic idea is sound. 
Nevertheless it fails to fill the place oc¬ 
cupied by the taper curves. It is still 
worth while to discover, if possible, im¬ 
proved methods of curve construction. 
MATHEMATICAL EXPRESSION OF 
TREE FORM 
A simple mathematical expression of 
form, and a generalized equation of tree 
curves has been sought by European 
foresters for many years, but none has 
proven entirely satisfactory (5). There 
have been no attempts of this kind 
in America except the recent modifi¬ 
cation of the Hojer formula, worked 
out by Behre (3) for western yellow 
pine in Idaho, which is of too recent 
introduction' to have proved its gen¬ 
eral usefulness as yet. The discovery 
of a simple general curve equation of 
this kind would go far in solving tree 
mensuration problems. In the search 
for such an equation, however, certain 
facts have been discovered which pave 
the way for a much simpler method of 
expressing tree form through taper 
curves. 
PROPOSED SUBORDINATE FORM 
QUOTIENT METHOD 
Tor Jonson, according to Claughton- 
Wallin, has proved that the taper of 
the trees of the same form class (form 
quotient) is independent of height with 
certain north European conifers. (The 
