
Tree shape Forest Measurement and Modelling. 

The diameter of a tree bole generally decreases or tapers from the base to the tip. The way in which this decrease occurs defines the bole form. This taper can occur at different rates and in different ways or shapes. An understanding of the form of the bole can allow:
There is a complex interaction between the bole form and the tree crown. Thus, any factor that influences the crown may also influence the bole form. Different parts of the bole grow at different rates as environmental and other factors affect the crown and the way photosynthates are distributed. 

Geometric shapes 
The taper of a geometric shape of regular outline can be described by the rate (k) and shape (b) in the general equation: where y denotes diameter or radius and x denotes the distance from the apex of the shape. Specific values of b correspond to common shapes:

Parts of a tree stem tend to approximate truncated parts of these common shapes. The base of the tree tends to be neiloid while the tip tends to be conoid. The main part of the bole tends to be paraboloid. The points of inflection between these shapes however are not constant. Paraboloid shapes can be further broken down into quadratic and cubic paraboloids. A quadratic paraboloid (b=1) would generate a straight line if height were plotted against radius squared, while a cubic paraboloid (b=0.66) would generate a straight line if height were plotted against radius cubed. Metzger proposed that a tree bole should be similar to a cubic paraboloid. He argued that the stem was a beam of uniform resistance to bending anchored at the base, and functioning as a lever arm. A horizontal force on such a beam would exert a strain that increased toward the point of anchorage, and the most economical shape for this beam would be a uniform taper similar to a truncated cubic paraboloid. Gray (1956) (quoted in Carron 1968) contended that the tree stem is not firmly anchored to the ground. A quadratic paraboloid shape would be more consistent with the mechanical needs imposed by this assumption. The above mechanical theories of stem form are one approach towards an explanation of tree shape. Two other theories relate tree bole shape to the need of the tree to tranport water or nutrients within the tree (water conducting theory and nutritional theory of stem form respectively). These theories are based on ideas that deal with the movement of liquids through pipes. The hormonal theory of stem form envisages that growth substances, originating in the crown, are distributed around and down the bole to control the activity of the cambium. These substances would reduce or enhance radial growth at specific locations on the bole and thus affect bole shape. 

Form factor 
A form factor is a summary of the overall stem shape. The volume of the stem is compared to the volume of a standard geometric solid of the same diameter at the base and total height. The most common form factor is the breast height form factor. The standard geometric shape for the breast height form factor is a cylinder of the same height as the stem and with a sectional area equal to the sectional area of the stem at breast height (i.e. basal area). The form factor is the ratio of the volume of the stem to the volume of the cylinder. Specific breast height form factors suggest general stem shapes:
If the appropriate breast height form factor for a tree of a given age, species and site can be easily determined, then the stem volume is easily calculated by multiplying the form factor by the tree height and basal area. 
Form quotient 
Form quotient is a ratio of the diameter at some point above breast height to the diameter at breast height. Again, it is a summary of the overall stem shape. The absolute form quotient was commonly used to group trees into form classes. This quotient is calculated by measuring the diameter at a height halfway between breast height and total tree height. This diameter is then divided by the diameter at breast height and expressed as a decimal. Absolute form quotients also suggest general stem shapes:
Lewis et al (1976) use the difference in diameter between 4.5 m or 7.5 m and 1.5 m to estimate the general shape of Pinus radiata in South Australian plantations. This value is used to improve the estimate of stem volume. 
Taper equation Taper model 
Taper equations are developed to predict the diameter at any point on the tree stem, i.e. model the stem profile. Ideally these predictions are unbiased and based on simple input variables like diameter at breast height and total tree height. The construction of these equations has been continuing for many years and has followed numerous approaches. Early researchers (e.g. Fries and Matern, 1965) used the newly available power of digital computers to fit complex polynomial equations. Often the tree bole was segmented into two or three parts and separate equations fitted to each part (e.g. Max and Burkhart, 1976). Other researchers (e.g. Goulding and Murray, 1975) emphasised the need for taper equations to produce estimates that were compatible with predictions of stem volume. Yet other approaches (e.g. Kozak 1988 and Bi 2000) attempt to develop a model that describes the continuous change in stem form from ground to tip. The variety of taper equation approaches attempt to overcome some general weaknesses:
For example, a cubic polynomial equation fitted against a subset of Pinus radiata height : diameter data from Crook (1998) explained 98% of the variation. The overall error in diameter prediction was close to zero, but there were localised errors in the stump (average 1 cm underestimate of diameter) and lower 10%  40% of the bole height (average 1 cm overestimate). These errors were balanced by corresponding errors further up the stem. However, the majority of stem volume and high value wood products tends to be in the lower third of the stem, so these local biases are significant. Grosenbaugh (1966) wrote: Many mensurationists have sought to discover a single, simple twovariable 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.Rivers (1995) found that localised biases in diameter prediction could cause significant biases when the predicted stem profile is used to estimate the size or value of products in a tree. In an experiment using over 100 trees, he found that the use of a taper equation resulted in a 10% overestimate of product volume and a 14% overestimate of product value  the taper equation was unbiased for diameter overall. A small bias in a diameter estimate may result in a large error in value if prediction of a product constraint (e.g. small end diameter of a log) is affected. Bi (2000) published the coefficients of a trigonometric variableform tape equation for 26 common Eucalyptus species. He conducted extensive tests on the localised biases of this model. 
[shape.htm] Revision: 6/1999 