Stand Density and Stocking Terminology Silviculturists are often interested in three related measures of stand density: absolute density, relative density and stocking. These measures can be used to describe a stand relative to some standard of comparison or to some condition that meets a silvicultural objective. Silvicultural decisions are often based on such measures of stand density, and the desired condition of the stand after treatment is usually described by these measures. Although the terms absolute density, relative density and stocking have not always been used consistently, some general conventions and definitions have been established (Walker, 1956; Bickford et al., 1957; Gingrich, 1964; Husch et al., 1982; Ernst and Knapp, 1985). Absolute density (or simply density in common usage) is a quantitative, objective measure of one or more physical characteristics of a forest stand expressed per unit area. Measures of absolute density are expressed quantitatively as a tree count, area, volume or mass. Ecologists usually use the term density to refer exclusively to the number of individuals per unit area. In forestry, however, the term can refer to any of several measures of site occupancy, including number of trees, basal area or volume per unit area. Measures of density are usually restricted to trees larger than some minimum size, usually expressed as a minimum dbh. Specifying this minimum size is important because absolute density usually differs with differences in the minimum measured tree size. Measures of relative density provide additional information by comparing an absolute density to a reference value. An example of a measure of relative density is the ratio of the number of trees of a given species per acre to the total number of trees per acre. Expressed as a percentage, this value has long been used by ecologists to define the relative density of a species within a specified area. Forest regeneration and growth can be greatly affected by relative stand density. Consequently, silviculturists have developed various methods of expressing relative density. Virtually all silvicultural definitions of relative density involve ratios. For example, Reineke’s (1933) stand density index provides a reference line describing the maximum number of trees per acre for stands of a given mean dbh (Fig. 6.1). The maximum number of trees decreases rapidly as the mean stand diameter increases. For any given stand, the observed number of trees and mean dbh can be used to compute the ratio (or percentage) of the number of the maximum (reference) number of trees indicated by the stand density index relation (Reineke, 1933; Schnur, 1937). Other common measures of relative oak stand density are based on tree–area ratios or stocking percent (Chisman and Schumacher, 1940; Gingrich 1967; Ernst and Knapp, 1985). In their application to oak forest types in North America, most measures of relative density are designed to compare one or more absolute measures of stand density to a standard. The standard is often based on an observed maximum absolute density for undisturbed natural stands at a comparable stage of development, but it may be based on other limits or reference conditions. For example, crown competition factor (CCF) estimates stand density relative to a minimum tree crown area per acre below which trees do not fully utilize available growing space (Krajicek et al., 1961). The method of French ‘normes’ compares the observed number of trees and the mean height of dominant trees to both a maximum density and the minimum number of trees necessary to maintain dbh growth below 2 mm (0.08 inch) per year, which by European standards is considered most desirable for veneer production (Oswald, 1982). Other measures of relative density that generally have not been applied to oak forests, but could be, include: Curtis’s (1982) relative density index (references observed basal area per acre to that of an undisturbed stand with the same quadratic mean diameter); Wilson’s (1946) relative spacing index (references the observed number of trees per acre to the number of trees in an undisturbed stand having the same dominant height); and Drew and Flewelling’s (1977) relative density index (references number of trees per unit area to the volume of the average tree). The latter method is analogous to the graphical format for expressing the -3/2 power rule discussed earlier. Comprehensive reviews of measures of relative density include those by Curtis (1970) and Stout and Larson (1988). Stocking is a subjective term used to describe the adequacy of any observed level of stand density with respect to a silvicultural goal (Bickford et al., 1957; Gingrich, 1964). The terms overstocked, understocked and fully stocked are used to describe stocking adequacy relative to a specified silvicultural goal. Accordingly, a stand may be overstocked (too dense) for one silvicultural objective and fully (i.e. appropriately) stocked for another, or may be overstocked at one age and understocked at another. In contrast to the term stocking, the term stocking per cent is a measure of relative density specifically associated with the Gingrich-style stocking diagram. This diagram combines measures of absolute and relative density into a single graphical format (Gingrich, 1967). Stocking per cent is a widely used measure of stand density in North American oak silviculture. It is based on the relation between tree size and associated growing space requirements discussed later in this chapter. The word stocking is often used incorrectly to refer to stocking per cent (a measure of relative density). This sometimes creates confusion because, as discussed later, full stocking is synonymous with complete utilization of growing space, which covers a wide range of stocking percentages on the Gingrich stocking diagram. Normal stocking is a term used to describe undisturbed even-aged stands that are at or near maximum density for their age. Normally stocked stands are characterized by a lack of gaps in the forest canopy and a relatively uniform spacing between stems. Basal area and cubic foot volume are at or near their maximum for a given stand age and site quality. Normally stocked stands (sometimes simply called normal stands) usually are identified subjectively based on these criteria. Observations of the number, basal area and volume of trees per acre in normally stocked stands across a wide range of stand age and site quality classes have been used to develop normal yield tables These tables specify the expected maximum basal area and maximum cubic foot volume for unmanaged stands of a given age and site class. In addition to their application to yield estimation, the tabulated values can be used as reference conditions to estimate the relative density of other stands. Maximum and minimum growing space There are limits to the amount of growing space a tree of a given bole diameter can occupy. Although this may seem self- evident, the concept is central to quantifying stand density and stocking per cent in oak stands. The actual amount of space that a tree occupies is difficult to measure because it includes crowns and roots that overlap in three dimensions with other trees. Fortunately, for many silvicultural purposes, a tree’s growing space can be adequately estimated as a circular area, or tree area, representing the crown. In this context, tree area is interpreted geometrically as a tree’s area of influence or potential influence concentric to the tree bole; it is also highly correlated with dbh. Estimates of the maximum area that a tree of a given dbh can occupy are usually developed from crown and dbh measurements of open-grown trees. In contrast, estimates of the minimum area that a tree requires are usually developed from measurements of tree diameters in normally stocked stands. Trees that are open-grown throughout their lives develop the largest crowns possible for their dbh and species. Consequently, open-grown trees have often been used to estimate the maximum area a tree of given species and dbh can occupy. There is a high correlation between bole diameter and crown area of open-grown trees. This relation has led to the development of equations for estimating the crown areas of open-grown trees from dbh for various oaks and associated species in several regions in the eastern United States. The results have shown that the relation between maximum crown width and bole diameter is often linear or nearly linear (Krajicek et al., 1961; Krajicek, 1967; Ek, 1974). An example is the maximum crown width equation applicable to oaks and hickories in the Central Hardwood Region, which is given by: CWmax = 3.12 + 1.829D [6.9] where CWmax is the estimated crown width (ft) of an open-grown upland oak or hickory, and D is tree dbh (inches) (Krajicek et al., 1961). Assuming tree crowns are circular, squaring both sides of Equation 6.9 and multiplying by Π/4 defines maximum crown area (CAmax) in relation to dbh so that: 2 CAmax = 7.645 + 8.965D + 2.627D [6.10] 2 CAmax therefore is the approximate circular crown area (ft in vertical projection) of an open-grown upland oak or hickory. Maximum crown width equations also have been derived for other species and regions (Table 6.1). An exponent in the diameter term of some equations indicates non-linearity in the relation. As in the derivation of Equation 6.10, equations in Table 6.1 can be similarly expressed as crown area. Graphical presentation of equations facilitates comparisons among species. For example, open-grown black walnut trees have larger crowns than oaks and hickories for a given diameter, whereas shortleaf pines have smaller crowns. The maximum crown width of sugar maple may be larger or smaller than that of oaks and hickories, depending on dbh (Fig.