## Beyond Yield Tables

In the twentieth century, yield table methodologies had become the standard for forestry applications and were essential part to any forestry curriculum. Considerable effort went into enriching the data sets and perfecting the basic concepts of yield tables for forestry production. For example, since the product of a forest plantation is not always wood but is often boards of a given dimension, allowing a forest to grow more wood (as predicted by a yield table) does not necessarily produce more boards. As another example, under certain spacing, the trunks of trees have different degrees of taper which also affect the lengths of standard boards to be sawn from the harvested wood. These sorts of complexities were determined and calibrated into the predictions from yield tables.

In the mid- to late-1950s several developments conspired to generate difficulties for yield table projections. The dependency on extensive long-term data sets to calibrate yield tables manifested itself as a problem. Some of the long data sets in Europe were initiated before the end of the so-called 'Little Ice Age' around the 1850s and thus were collected over intervals of significant climate applications. The warmer and often dryer conditions that prevailed in the decades of 1930s and the 1950s represented a similar problem for shorter calibrations of other yield tables. The fundamental assumption in yield tables, that the expected height of a canopy tree in a stand of a reference age could be used as a constant to encapsulate the quality of a site for growing trees, was undermined by these climate variations.

At the same time and particularly in the US, foresters were experimenting with developments that would change the growth rates of trees - using genetic selection to develop faster-growing commercial tree species or fertilizing forest for faster growth. Increased air pollutants and changes in the chemistry of rainfall ('acid rain') were seen as potentially slowing tree growth. The possibility of metrification in the US meant that the complex calculations to calibrate merchantable-dimension boards in English units would need to be recalibrated for metric units. In the western states of the US, the end of harvest of previously uncut forests of Douglas fir (Pseudotsuga menziesii) and the interests in developing plantation forestry for this species poised the immediate question of how does one develop a yield table for a condition for which there is no long-term data record. Japanese foresters faced a similar problem with their plantation forestry practice at home and in foreign countries. In tropical and subtropical locations, where the forests have relatively little softwood timber, planting of exotic North American species, notably Monterey pine, Pinus radiata, gave local foresters a need to develop planting strategies with no data to calibrate yield tables. Worldwide, the realization that forests do more than just produce wood called for a simultaneous management of forests for water quality, wildlife, flood control, landslide protection, and recreation, all implied a broader approach than the wood production emphasis of yield table forestry. This soon was seen as need for a better capability to manage complex, natural forests with a mixture of tree ages and species. There was a considerable impetus to develop new approaches to projecting forest dynamics beyond the yield table approach.

In the late 1950s and early 1960s, the need for forest prediction beyond the capabilities of empirical yield tables led to a proliferation of forest modeling approaches, notably in the US and Japan. This modeling revolution in the late 1960s and 1970s occurred independently at different locations with different emphases on theory development, computer simulation, and advanced statistics. These developments, intended to re-analyze the intensive data archives behind the yield tables and extend insights of a century of forestry empirical-model predictions, coincided with a fascination with the potential for high-speed computers. Many of the resulting models depend strongly on digital computation and have become more widespread as the power of computers has increased over the past 50 years.

Each of these different foci on the modeling of forests contributed different points of view as to how to predict the dynamics of forest ecosystems and are discussed in the sections below. In the case of forestry models, three significant intellectual centers for this development were

1. a Japanese group of forest scientists who emphasized the theoretical underpinnings for forest dynamics;

2. one US center focused on the statistics needed to parametrize coupled differential equations for biomass, number of trees, and size of the average tree in forest plantations. This group was strongly oriented to understand the dynamics of Loblolly pine (Pinus teada) plantations;

3. a second US group developing digital computer simulation models that incorporated the three-dimensional geometry ofinteractions among growing trees, initially emphasizing Douglas fir and spruce forests. The approach was to develop what is referred to today as individual-based or agent-based models.

Each of these groups produced different insights on the population dynamics of forests and has evolved in relative independence until the present day.

At about the same time that new alternatives to yield tables were being developed in forestry, ecologists were developing a parallel interest in the use of dynamic models to predict forest dynamics. This culminated in a major international research program called the International Biological Programme (IBP) intended to foster a comparative understanding of ecosystem dynamics across the globe. The focus was on the productivity and element processing of natural ecosystems and constitutes a fourth independent thrust to develop forest ecosystem models. The models developed in this program tended to emphasize fundamental processes (productivity, decomposition, element uptake by ecosystems). These models represent a relatively significant departure for the forestry modeling tradition in assuming these processes can be represented over a homogeneous forested area. The IBP models are in many cases the predecessors of ecological models currently applied to compute the carbon dynamics to predict the responses of the global carbon cycle.

Japanese Theoretical Approaches to Forest Population Modeling

While the German traditional approach to yield table development were based on the normalization of data based on tree heights, Japanese foresters focused on the statistical interrelations among all of the dimensions of trees. They developed a geometrical theory for the dynamics of forest stands using allometric equations. These equations express one measurable dimension of an organism as another dimension raised to a fractional power times a constant (x1 = cxZ, where x1 and x2 are different measurements of a tree and c and z are constants). At the beginning of the twentieth century, the British mathematical ecologist, D'Arcy Wentworth Thompson, had written extensively about these and other more complex underlying mathematical formulations describing the shapes of plants and animals. The Japanese compilation of these data for trees was extensive. Significantly, K. Shinozaki and his colleagues developed a basic theory (in 1964) to explain some of these regularities in form.

Shinozaki's 'pipe model' posited that a tree is a great collection of pipes - notably the xylem cells, tubes that transmit water from roots through the trunks and branches to the leaves. In the pipe model, the cross-sectional area of the tree's transport systems (the 'pipes') should be conserved at different levels of the tree. A given area of leaves required a certain cross-sectional area of xylem tubes through all the parts of the tree that supplied it. As a tree grows taller, more and more length of pipes is needed to provide the same cross-sectional area. Thus, the cost to the tree of supporting a given area of canopy leaves (requiring a given cross-sectional area of pipes) increases as the tree grows. The exponent in an allometric equation typically has values near one-third when relating a linear dimension such as tree's diameter to volume-related measurements such as tree biomass and a value near two-thirds when relating area measurements such as cross-sectional area of a tree trunk to tree mass.

In 1963, the Japanese forester, K. Yoda, and several of his colleagues extended the classic allometric concept for individual organisms to entire forest stands. The motivation was to investigate one of the basic premises in yield tables - the regularities in the average size of trees in

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Tree density (ha-1)

Figure 2 An example of a -3/2 law stand-thinning concept. Forest stands planted at different densities converge to a limit described by a line with a slope of -3/2 (or -1.5). Data are for Loblolly pine plantations with different initial planting densities (79 trees ha-1; 149 trees ha-1; 233 trees ha-1; 430 trees ha-1; 1141 trees ha-1). The points represent stand inventory starting in 1935 when the experimental forest was established at the Duke University Experimental Forest near Durham, New Hampshire. As trees increase in size, numbers are reduced in these even-aged stands. Reproduced from Peet RK and Christensen NL (1987) Competition and tree death. Bioscience 37: 586-595, with permission from American Institute of Biological Science.

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Tree density (ha-1)

Figure 2 An example of a -3/2 law stand-thinning concept. Forest stands planted at different densities converge to a limit described by a line with a slope of -3/2 (or -1.5). Data are for Loblolly pine plantations with different initial planting densities (79 trees ha-1; 149 trees ha-1; 233 trees ha-1; 430 trees ha-1; 1141 trees ha-1). The points represent stand inventory starting in 1935 when the experimental forest was established at the Duke University Experimental Forest near Durham, New Hampshire. As trees increase in size, numbers are reduced in these even-aged stands. Reproduced from Peet RK and Christensen NL (1987) Competition and tree death. Bioscience 37: 586-595, with permission from American Institute of Biological Science.

even-aged forest stands as a function of the density of trees. In self-thinning stands (stands in which some trees die from competition as other, more vigorous trees grow), the average biomass of a tree is a constant times the density of trees raised to the -3/2 power (Figure 2). This -3/2 power law or self-thinning law has been found in a wide variety of ecosystems in which organisms (corals, grasses, trees) compete for space. Its application is restricted to even-aged systems that have been initiated with a competing cohort of individuals.

Models Coupling Mass and Numbers Equations for Forest Dynamics

At about the same time that foresters in Japan were developing their theoretical concepts, J. L. Clutter and a group of colleagues and students at the University of Georgia were developing an alternative approach with the same underlying concept as that of the Japanese-connecting the size of an average tree to the number of trees in an even-aged forest stand. The Georgian approach was to conceive stand dynamics as being modeled by two differential or difference equations, one equation for the change of numbers over time and the other for the change in the size of the average tree over time. These two equations were coupled to one another by statistically estimated terms. For example, if the weaker trees that die as the stand trees are smaller than average, then tree death increases the average tree size and a part of the average-size equation would have a term that expresses this effect. The terms in each equation that coupled numbers to size (and vice versa) were estimated statistically from the same sort of data that is used in yield tables in some cases. In other cases, the need for information on how to couple the number and mass equations suggested new experiments or new data collections. The advantage of the method was that if genetics or forest fertilization made the trees grow faster, this effect could be directly added to the average tree mass equation. The estimation of the parameters in the coupled equations could be improved as more field data were obtained. Thus, the models were expected to improve over time.

Today, this approach and its descendents dominate plantation forestry, particularly for Loblolly pine (Pinus taeda), the principal commercial tree species in the US. The method works best on forests in which the trees do not vary greatly from one another in size (hence, the average tree does not have large associated variability).

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