BACKGROUND I I
This study does not pretend to cover the case of trees which, owing to old
age, decay, or unrestricted branch development, are quite irregular in
shape, but deals with forest-grown specimens, the stems of which are
normally branchless for an appreciable proportion of their total length or
at least are reasonably symmetrical.
(b) 'Form' and 'Taper' defined
Within this narrower field of discussion, the terms 'form' and 'taper' are
frequently used indiscriminately to express the same idea. This has been a
major cause of confusion, and for the purposes of this paper these two
terms will be separately defined with reference to three of the geometrical
solids which narrow in diameter from the base upwards in a regular
fashion, to which it has become customary to compare the main stem of
a tree.
(i) 'Form'
The particular fashion in which such a solid narrows in diameter so as to
produce a characteristic shape depends on the power index of d in the
formulae for the various curves defining the diameter/height profile of such
solids, thus:
for a cone h = hi d
for a (quadratic) paraboloidt h = h2 d2
for a cubic paraboloid h = k3 d3
where d is diameter at distance h from the vertex of the curve, and h represents
a value which can vary from case to case.
By 'form' in this paper is meant the shape of a solid, the diameter/height
curve of which is determined by the power index of d.
(ii) 'Taper'
Depending on whether the rate of narrowing in diameter with respect to
increase in height is slow or rapid, any solid of the above types is relatively
tall and slender or short and stout. A high value indicates a slow rate of
narrowing and a low value indicates a rapid rate of narrowing in all cases.
By 'taper' in this paper is meant the rate of narrowing in diameter in relation
to increase in height of a given 'shape' or 'form'.
(iii) Diagrammatic distinction between 'form' and 'taper'
Full appreciation of this distinction is so important to a proper understanding
of the subject of stem form that the text above is amplified by
simple diagrams. Fig. 2 (a) portrays conical, paraboloid, and cubical paraboloid
forms. Each of these bodies is shown with slow, intermediate, and
rapid taper.
t Unless qualified by a prefix, e.g. 'cubic', the term paraboloid in this paper has the
usual mathematical meaning, that is a paraboloid of revolution generated by a parabola
and often referred to as a quadratic paraboloid.