Stem form and taper ©

In forest mensuration stem form (synonymous with stem shape), is determined by the way the stem tapers, i.e. by the decrease in diameter from base to tip.

Taper of a solid body of regular outline is governed by the values of 'rate' and 'shape' in the expression:

diameter^2 = rateheight^shape

where shape determines the way the body tapers (i.e. form or shape) and rate the rate of taper within that form.

If we knew more about the factors determining stem form, we might be in a better position to control it: a cylindrical stem would be an obvious advantage in utilisation! Certainly species, genotype, age, competition, site (especially wind exposure), silvicultural treatment and size and structure of the crown are all important. Of these factors, the crown, particularly crown length, plays a decisive role in determining stem form.

Some general patterns are:

Tree situation Ring area growth Ring width
Free growing trees, predominants or vigorous trees growing in gaps Crowns strongly developed Increases basipetally Increases basipetally
Stand grown trees not overtopped but side growth of crown hindered e.g. codominants Approx. equal growth along length of stem Decreases basipetally
Overtopped stand grown trees, e.g intermediate and suppressed trees Decreases basipetally Decreases basipetally
These patterns suggest that trees tend to become more cylindrical with increasing stand density and decreasing crown class. This was confirmed for radiata pine by Siemon (1973). He found that stems became more cylindrical with time in unthinned stands, though taper altered little or not at all in the lower bole, whereas in heavily thinned stands, taper increased with time but shape remained the same.

Although crown size and development definitely determine stem growth, many variables (age, site, location in the stand, etc.) intercede to further determine how this growth is distributed over the stem.

Because of the downward increase of ring area increment within the crown, a point or region of maximum growth is reached in the general vicinity of the live crown base or, perhaps more accurately, at the position of maximum branch development. This maximum continually shifts upwards as the tree matures and the crown recedes. In favourable growth years, however, the maximum temporarily shifts downwards. On the other hand, in unfavourable years, movement upward is temporarily accelerated.

A pronounced region of minimum ring area growth is often encountered on the branch free bole. In young stems, it occurs near the ground shifting upwards towards the crown base in dry years and downwards towards the stem base in wet years, i.e. it varies with the prevailing growth conditions and parallels the region of maximum area growth near the crown base.

Regardless of the ring width pattern along the stem, a relatively high degree of taper is always found within the crown. This is because of the progressive increase in branch numbers downward from the apex and the cumulative contribution of these branches to stem growth.

Cultural measures such as thinning and pruning can markedly alter the incremental growth pattern of individual trees by artificially creating new growth conditions. Thinning converts a stand-grown tree to a simulated open-grown condition, and growth will shift downward on the stem in response to the increased crown size and exposure. Pruning, on the other hand, converts an open-grown or large-crowned tree to a simulated stand-grown condition, and growth will shift upward on the stem in accordance with the new crown position. Again, these responses are not invariable and a change in form often fails to occur following a silvicultural operation. The lack of response in many instances, however, can be traced to the prior condition of the trees, the experimental techniques, or more frequently, to the analysis and interpretation of the results.

Inheritance must also be taken into consideration, and it must constantly be born in mind that all trees are predisposed to assume a certain form. Environmental influences and cultural practices can only modify the basic tree form predestined by heredity.

Points of inflection along the stem present a major problem in specifying stem form. Besides the two major points, one in the butt region and the other somewhere in or close to the crown base, there are many minor points of inflection along the bole. In other words, tree form is complex. This makes it difficult to specify the profile of a single stem, let alone to specify a profile to represent all stems. Nevertheless, researchers have attempted to do this through two broad approaches:

Theoretical approaches (growth processes and tree form)

Various theories, called stem form theories, have been proposed to explain the variations in the way trees accumulate woody material with time. The four main theories are:

Support for both the Nutritional and Water Conducting theories has waned in recent decades.

Mechanistic theory

The Mechanistic Theory has received the greatest acceptance to date. It is suited to mathematical analysis and has found applications in practical forestry. Schwendener introduced the theory in 1874, but it was Metzger in a series of classical papers who developed Schwendener's tenets into true mechanistic laws and applied them to silvicultural practices.

Metzger proposed that the stem was a beam of uniform resistance to bending, anchored at the base and functioning as a lever arm. A horizontal force will exert a strain on the beam that increases toward the point of anchorage, and if the beam is composed of homogeneous material the most economical shape would be a beam of uniform taper. Metzger demonstrated that this taper approximates the dimensions of a truncated cubic paraboloid (height against diameter^3 is linear). He confirmed his theory for many stems, particularly conifers.

Gray (1956), an Australian forester, questioned Metzger's assumption that the tree stem is anchored firmly to the ground, and contended that stem form conforms more closely to the dimensions of a quadratic paraboloid (height against diameter^2 is linear). A stem of this shape would be consistent with the mechanical requirements of a tree, not only to horizontal wind pressure but to other forces acting on the stem.

There have been many attempts to prove and disprove the theory (e.g. Larson, B. 1963. For. Sci. Monogr. No. 5). Generally, it is accepted that the theory provides a realistic interpretation of tree form with respect to strength requirements and support function. Though the theory has survived experimental study reasonably well, it does not provide an adequate physiological explanation of the observed facts.

Hormonal theory

The Hormonal theory envisages growth substances originating in the crown regulating the distribution of radial growth on the stem by controlling the activity of the cambium.

It provides a physiological explanation of how a tree grows and why trees differ in the way they do, but it does not specify the particular shapes trees may have under varying circumstances.

The hormonal theory offers the most promising approach to the stem form problem. It provides a physiological basis for the nutritional as well as the functional theories of stem form.

Empirical approach (stem profiles and tree form)

Empirical approaches describe tree shape and taper without incorporating a theoretical basis of why trees assume the described shape. Ideally these approaches will summarise the description to a single statistic or simple equation. The best known empirical descriptions of tree form are the Form Factor and the Form Quotient.

Form factor

In general terms, the form factor of a tree is the ratio of its volume to the volume of a specified geometric solid of similar basal diameter and height.

Most commonly, the form factor of trees is based on a cylinder. Thus, the product of tree basal area, tree height and cylindrical form factor should give tree volume.

The stimulus for the determination of form factors was the recognition of the strong resemblance of a tree stem to standard geometric solids. The hope was that ratios would be established which could be used to convert easily computed volumes of standard solids into tree volumes.

Several types of form factor exist:

Because form factors are based on volumes (or diameter measurement is needed near the top of the tree as with Pressler), they can not be measured directly or easily on standing trees. For this reason, attempts were made to find a correlation between form factor and some index which could more conveniently be measured on a stem. Such a correlation was found with Form Quotient.

Form quotient

Form quotient is a ratio of diameter at some point above breast height to diameter at breast height.

As with form factor there are several types of form quotient, the most common being absolute form quotient which is the ratio of the diameter at half height above breast height to dbh. This is commonly written as d5/d0 and, conventionally, is expressed as a decimal, e.g. 0.70. Form quotient values are grouped into form classes, e.g. Form Class 70 comprises trees with F.Q.s from 0.675 to 0.724.

Absolute form quotient (and classes):

Tor Jonson was the most famous of a number of workers who have investigated the empirical evidence of tree shape. Based on numerous measurements at percentile points on west European conifers and, he proposed that the profile of trees above breast height could be described by the formula:
dl / d = C*log[(c+l-2.5)/c]
where dl = the diameter at a percentile distance l from the tip (l being expressed as a percent of height above breast height), d denotes d.b.h. and C,c denote constants varying with form class.

Jonson used this formula to compile tables giving, for various form classes, tree diameters at percentile heights as percent of dbh. Thus, the profile of a stem is defined and so, automatically, is its form factor.

Following Jonson, other workers (Wright, Behre, etc.) investigated other stem profile equations but none was found to be entirely satisfactory.

Gray developed his hypothesis of stem form based on empirical evidence (from numerous sample trees covering a wide range of species) that over a large part of a coniferous stem, height above ground is directly proportional to the square of diameter at that point. The practical application of his hypothesis is the taper line method of volume estimation.

Taper equations

With the advent of powerful computers, all sorts of polynomial curves have been fitted to the profile of individual trees. For example Fries and Matern (1965) fitted a polynomial up to 58th power. Kozak et al (1969) fitted a quadratic (polynomial to the power of 2) and found it acceptable.

Most are of these equations empirical and they do not cater adequately for the variability found in any one population. The nature of the stem profile is compound and it would seem that it can only be described adequately by separate expressions for the top section, main bole and buttswell region. In addition, the sections of stem described by these separate expressions vary with species, age, size, stand condition and environment.

Grosenbaugh (1966), following a comprehensive study of stem form, writes (Forest. Chronicle 42: 444-57):

Many mensurationists have sought to discover a single, simple two-variable function involving only a few parameters which could be used to specify the entire tree profile. Unfortunately, trees seem capable of assuming an infinite variety of shapes, and polynomials (or quotients of polynomials) with degree at least two greater than the observed number of inflections are needed to specify variously inflected forms.

Furthermore, coefficients would vary from tree to tree in ways that could only be known after each tree has been completely measured. Thus, explicit analytic definition (of the tree profile) requires considerable computational effort, yet lacks generality......

Each tree must be regarded as an individual that must be completely measured, or else as a member of a definite population whose average form (profile) can only be estimated by complete measurement of other members of the population selected according to a valid sampling plan.... Hence, polynomial analysis may rationalise observed variation (in the stem profile) after measurement, but it does not promise more efficient estimation procedures."

Thus, it appears unwise to derive complicated relationships to characterise the tree profile. This notion is corroborated by several researchers who, after studying various multivariate analysis techniques, concluded that the use of simple functions, sorting, and graphical methods is adequate for many uses in operations and research.

Tree shape in Australian forests

Absolute form quotients for plantation conifers in Australia range from 0.6 to 0.8. Stems with a FQ of approx. 0.8 also give a reasonable fit to Metzger's cubic paraboloid since the FQ of a cubic paraboloid is 0.8. Similarly, stems with a FQ of approx. 0.7 give a reasonable fit to Gray's 2 paraboloid since the FQ of a 2 paraboloid is 0.7.

A majority of coniferous stems in Australia have a FQ of about 0.7 so there is considerable evidence for the generality of Gray's hypothesis. The hypothesis also holds well for most, but not all, native broadleaf species from above the buttswell region to crown break.

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Cris.Brack@anu.edu.au
Tue, 7 Jan. 1997