Stand density
Forest Measurement and Modelling.

Broadly, we may define stand density as:
a quantitative measure of tree cover on an area, i.e., the amount of tree material per unit area or space.

Many foresters use the terms stand density and stocking synonymously. Conventionally, measures of stand density are accepted as being absolute and unaffected by management objectives. In contrast, stocking historically involves comparison of a given stand with some established norm, e.g., one established with a particular purpose of management in mind. Thus, stands are described as overstocked, 50% stocked, understocked, etc.

Because they are absolute, measures of stand density are more precise and more useful in analysis and estimation of forest growth and yield than stocking. Stand density is important in forestry because, within limits, the more growing space made available to a tree, the less competition it will face and the faster it will grow. Thus, an important role of the forester is to regulate stand density through initial planting spacing, thinning and other silvicultural practices.

Estimates of stand density are made to express the degree to which the growing space available for tree growth is utilised. Thus, stand density is a function of three elements:

  1. Number of trees - which is readily determined by counting.

  2. Tree size - which involves a number of factors, e.g.:-
    • Stem - characterised by diameter, height and taper
    • Crown - characterised by spread and depth
    • Root - characterised by spread and depth (both difficult to measure).

  3. Spatial distribution on the ground - which is not readily determined. Generally, a square or triangular spacing is assumed.

Drew and Flewelling (1977, 1979 - For. Sci. 23 and 25) define maximum density:

as the density at which a stand undergoes substantial and continuing mortality induced by competition.
In effect, this implies a natural barrier beyond which stands may not grow. Such a barrier is a useful base for planning thinning strategies. Applications of this barrier include the 3/2 power law of thinning or 'self-thinning' line. This line, relates the logarithm of the size (diameter, volume, weight) of the mean tree or stand to the logarithm of the number of trees per unit area, has a negative slope of -1.5 (-3/2).

Changes in density may affect both the timber yield from a site and the sizes of the individual trees on it. The effects can be measured in either absolute or relative terms. Because efficient management of a forest necessitates careful regulation of stand density, there is a need to describe it quantitatively. This is done by the use of what are termed stand density indices.

Stand Density Index In choosing a stand density index, several points should be kept in mind:

  • The variable used as the index should be simple and easily measured and applied in practice.

  • The index should imply the same thing to all people, and the different amounts of tree material they are variously interested in (tree biomass, foliage weight, sawlog content, etc.) should all be derivable from the index.

  • The index should be independent of age and site. Stratification of stands based on density as well as age and site will not improve volume estimates if the index of density is correlated with age and site.

  • Variation in the index of density of a stand should be reflected in a difference in growth behaviour. It should also reflect differences in growth behaviour between stands.

  • The index should be a variable which lends itself to forward projection, e.g. future yield is likely to be affected by the density at a future time which will be a reflection of the present density.

  • Ideally, the index should be applicable to any kind of stand, even-aged or uneven-aged, single or mixed species.

Many indices of density have been proposed and used by foresters (refer West, 1983) but not one meets all the requirements listed above. A number of different measures of density may be necessary to cover the wide range of conditions encountered in practice.

The measures of stand density fall into two classes - biological and practical.

  • Biological measures: These aim to describe the stand and reflect its growth behaviour.

  • Practical measures: These are those used as a basis of for regulating stand density, e.g. indices derived essentially for the practical purpose of implementing a thinning schedule.


The bases of the various indices of stand density used by foresters fall into four categories:

Number of trees

The number of trees on a defined area may be a satisfactory index of density if tree size is uniform or differences in size can be ignored, e.g., young stands following establishment where the main concern is whether or not the area is sufficiently stocked and whether refilling is necessary. Stocking here implies 'stems per unit area' and is a qualitative expression of the adequacy of tree cover on an area. Thus we use the relative terms 'under-stocked', 'fully stocked', 'over stocked' in describing a young forest stand. Stocking may also be expressed as a percentage of a pre-established norm, i.e., as a stocking factor.

Stocking is a practical index of density for regulating stands of a specified species, age and site where the desirable number of trees per unit area at a given stage of development is known from past experience or research. Generally, however, number of trees must be combined with tree size to be a satisfactory measure of density.

Number of trees and dbhob

Number of trees and dbh are incorporated in four indices, the first being the stand density index of Reineke (1933). Ferguson and Leech (1976) have shown that this index involves a source of bias which could lead to biased estimates of density. They also show it is analagous to stand basal area (see below). Reineke's index is extensively used in the U.S.A. but has found little application in Australia.

Stand basal area is another index incorporating number of trees and dbhob. The basal area of a stand of a given age varies with species for a given site and with site for a given species. However, for certain species, the basal area of stands on particular sites may be reasonably constant over a considerable period of development of the stand particularly towards maturity. Under this condition, stand basal area is a good measure of the maximum occupancy of the site and thus of stand density.

Stand basal area is widely used in the management of even-aged stands for a number of reasons, viz. it is a practical index of stand density; it is easily measured; it is the natural base for deriving stand volume; and volume increment and basal area increment are usually well correlated. Another index in this category is the crown competition factor (CCF). Leech (1984) demonstrated that crown width (CW) in open grown P. radiata in South Australia is linearly related to tree dbhob (d) viz.,

CW  =  0.7544  +  0.2073 d
      (0.0848)   (0.0032)

Leech used this model to calculate the CCF of P. radiata stands. Given an area of A hectares with a total stocking of N trees:

 CCF = 1/A [ 0.004 470N + 0.002 456 Sum(d) + 0.000 337 5 Sum(d^2) ] 

Leech suggests that the CCF should be a useful variable in growth and yield studies of radiata pine for it is independent of age and site.

Note that number of trees in relation to mean tree dbhob has often been used as a 'rule of thumb' method of thinning, e.g.,

Required spacing between trees (m) = 0.1 d (cm) +/- k.

Crown cover is another index incorporating number of trees and dbhob (through the crown diameter/dbhob relationship). The vigour of tree crowns is partly related to aerial growing space so crowns might be expected to reflect the density of a stand. Another is crown closure (also termed canopy closure), which is the ratio of the projected horizontal crown area to the total horizontal ground area, is used as an independent variable in aerial stand volume tables. Strictly speaking, it is an index of 'area occupation' rather than stand density.

In even-aged stands, crown closure may be proportional to basal area/ha. This relationship has led to the development of indices between estimates of crown closure obtained from aerial photographs and basal area. The value of crown closure as a variable depends on how well variation in stand volume is correlated with it: root space may be more important than crown space in determining stand growth!

Number of trees and height (Hart's Index)

Indices of this type are of the practical kind for implementing schedules for regulating stand density, e.g. height/spacing ratios (HSRs). They are mainly used in even-aged stands of known age and site.
HSR  =  H/S, thus S = H/HSR                        
where	HSR is the height/spacing ratio;
	H is some index of stand height 
        (mean height, predominant height, top height, etc.);
and	S is the average spacing between trees. 

It incorporates the number of trees per unit area and may be based on a triangular spacing (Holland) or square spacing (UK).

H should effectively reflect the site/age relationship because as site quality and age increase, spacing should increase. For this reason, predominant height (or top height) is the commonly used stand height index.

Number of trees, dbhob and height

This category contains two indices, stand bole area and stand volume.

Stand Bole Area:

The bole area of a tree is its surface area underbark. It represents the base for potential increase in tree volume. Wood increment over a period is superimposed on bole area at the beginning of the period.

The total bole area for all trees in a stand represents the base for potential increase in stand volume, therefore tree and stand bole areas should be variables of particular interest to foresters. Surprisingly, little attention has been given to them.

Most work on bole area has been done by Lexen (J. For. 41:883-5, 1943). Lexen's argument is that stand volume only provides a measure of wood capital and not of the potential of the stand to grow. He suggested that estimates of stand volume should be supplemented by estimates of stand bole area.

For the individual tree, bole area can be estimated by substituting girth for sectional area in Huber's or Smalian's formulae, or plotting girth against height on rectangular co-ordinate paper, and square counting. Lexen compiled bole area tables for P. ponderosa based on dbhob and height. He found that bole area of the individual stem could be approximated by applying the formula:

B = k dh
where   d is dbhob(ins)
h is total height (ft)
and k = 1/7 or 0.143 (= 0.0172 metric.  Metric = Imperial x 12/100).
Carron (1968) investigated bole area in P. radiata covering all sites and ages in Uriarra Forest A.C.T. and found k varied from 0.137 (0.0164 metric) to 0.154 (0.0185 metric) and was correlated with dbhob but was unaffected by height class within a dbhob class (Table 9-1). Overall, Carron found variation in 'k' was so small that he suggested a useful approximation to stand bole area (SBo.A) could be obtained by applying the formula:
SBo.A (m2) = Stand mean d (cm) x Stand mean h (m) x No. of trees x 0.0172
Stand bole area is a function of diameter, height, stocking and tree form and is thus a measure of the amount of stand in a physical sense. Also, as it is the base for potential increase in stand volume and describes a surface that absorbs short wave radiation from the atmosphere and re-radiates long wave radiation onto surrounding areas, it has particular biological and ecological significance. Thus, being a measure of the amount of stand in both the physical and biological senses, stand bole area should be an effective index of density - but the evidence is conflicting (refer p. 141 Carron).

Stand Volume:

The volume of trees/unit area is an expression of density. Ultimately density must be expressed in terms of volume for economic evaluation of forest management strategies. However, volume is difficult to measure and it is not independent of age and site.

Other Measures of Stand Density

Other measures of density (e.g., tree area ratio) which are claimed to be independent of age and site are described in the literature (e.g. Husch, Miller and Beers, 1982. "Forest Mensuration", Chp. 17). These warrant close examination by any researcher involved in investigating relationships between stand variables and tree growth.

In recent years many attempts have been made to describe such relationships by equations. Many indices of density have been tested in these equations, the efficiency of a variable as a measure of density being judged mathematically by its contribution to the correlation. Almost invariably, basal area (G) has emerged as the most satisfactory variable. West (1983), in a detailed study of 17 measures of stand density in even-aged regrowth eucalypt forest in southern Tasmania, concluded that the more complex measures were no better than the simpler ones, e.g., G.

Point Density The previously described measures of stand density are usually employed to determine the average density of a stand. Somewhat more specific measures of density have been developed to describe the degree of competition at a given point in a stand. These measures of what is termed Point Density are useful in silviculture and ecology to evaluate the effects of competition upon single trees in a stand, e.g. the correlation between the growth of a tree and density around the tree, the correlation between establishment of natural regeneration and point density, and the selection of plus trees for research in forest tree improvement.

The 'angle summation method' of Spurr (For. Sci. 8: 85-96, 1962) is an example of one technique giving a measure of point density. The method involves choosing a point or tree upon which we wish to determine the degree of competition from surrounding trees. Using the basic theory of Bitterlich (angle count sampling) each competing tree is imagined to be 'borderline' from the chosen point or tree and, thus, to have a specific basal area factor. The point density measure is obtained by appropriately summing a series of basal area per hectare estimates made using these trees.


Point density (m2/ha) = [0.25(0.5(d1/l1)2 + 1.5((d2/l2)2  + .. 
                             +(n-0.5)(dn/ln)2 )]/n    

where: 	di is diameter in cm of tree i; (i = 1, ...., n).
	li is the distance in metres of tree i from the sample point or subject tree;
	n  is the number of competing trees measured.

[temp.htm] Revision: 6/1999