| Summary: | A knowledge of the form of tree stems, and of the manner in which such stems vary in taper, are of importance to the forester in the determination of volume and in the construction of volume tables. At present there are two theories relating to stem-form, namely those of Metzger and Gray, and several formulae for constructing stem-profiles.
The theory of Metzger is the oldest and most well known. He claimed that the form of the forest tree stem is not fortuitous but depends on certain forces acting on it, of which wind is the most important. Metzger described the tree stem as a "beam of uniform resistance" which, according to the laws of statics, is a cubic paraboloid. More recently Gray has disagreed with this theory, arguing that because the base of the beam is not fixed in a solid stratum, as Metzger supposed, the cubic paraboloid would in fact be too strong and therefore uneconomical. He claimed that the most economical stem form is that of a quadratic paraboloid.
The quadratic paraboloid was tested for a number of trees from the University Research Forest and in each case was found to be closely correlated with the actual stem profile, except at the butt due to butt swell, and at the top where the stem resembled a cone. The cubic paraboloid was found to give a good fit in the lower part of the stem but under-estimated the diameter in the upper part of the stem.
An American investigator, C. E. Behre, described the stem profile as a hyperbola. Although no theory has been published as to why this should be the case, the formula derived by Behre has gained wide acceptance in North America and many other parts of the world. The formula, in theory at least, gives a perfect fit to a cylinder or a cone and also good fits to the major parts of quadratic and cubic paraboloids and intermediate forms of stem profile. When used in this study, Behre's formula was found to give a slightly better fit than the quadratic paraboloid.
The formula describing the stem profile of a quadratic paraboloid, called the "taper-line" by Gray, is of the form D² = a - bH, where D is the diameter of the stem at a height H, "a" is the regression constant, and "b" is the regression coefficient. The regression coefficient is an index of the slope of the taper-line, and therefore of taper, and can be used to trace the pattern of taper variation with various factors which are thought to be related to taper. By using multiple regression techniques it is possible to reduce these factors to one or two. In one such test on ten Douglas fir trees from a mixed fir, hemlock, and cedar stand, average age about 65 years, it was found that age and site index were not important but that diameter at breast height, D₄.₅²/Ht, and total height, Ht, were. The final regression obtained was b = 0.0242 + 0.998840 D₄.₅²/Ht. A similar regression was obtained for "a", the regression constant, and tables of values of "a" and of "b" were constructed. From these tables it is possible to derive the taper-line for any tree of known d.b.h. and height. By this means it is possible to calculate the volume of standing trees.
The main conclusions reached in this thesis are that the form of forest tree stems is complex and cannot be ascribed readily to either of the known theories; the quadratic paraboloid gives a sufficiently good fit over the main part of the stem (between about 15 and 80 per cent of the total height) that it can be used as a basis for studying taper variation; and finally, that the amount of taper present in a stem varies directly as the square of diameter at breast height and inversely as the total height. The amount of taper appears to increase throughout the life of the tree as long as the tree remains in the dominant crown class. As soon as the tree passes into the codominant class the rate of increase in taper falls off and when the tree passes into the intermediate and suppressed classes the amount of taper decreases with increasing age. The form of a tree grown in the open, that is apparently free from competition from surrounding trees, resembles a cone, or a neiloid in some cases, and not the quadratic paraboloid of a close-grown or forest tree. === Forestry, Faculty of === Graduate
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