The second approach recognizes that most forest rate data are periodic in nature and, 

 therefore, an appropriate way to model them is by using difference equations. An example of 

 an uneven-aged, whole-stand simulator using this approach is Ek's (1974) mixed species, north- 

 ern hardwood simulator that characterizes the stand by number of trees in 2-inch diameter 

 classes. In this simulator, the number of trees in each diameter class at the end of a growth 

 period is a linear combination of ingrowth, upgrowth, and mortality. Ingrowth is the number 

 of trees growing into the smallest diameter class in a growth period, and Ek modeled it as a 

 nonlinear function of total stand basal area and total number of trees. Upgrowth is the 

 number of trees growing out of a specified diameter class in a growth period, and it was 

 modeled as a nonlinear function of number of trees and basal area in the diameter class, total 

 number of trees and basal area in the stand, and site index. Mortality is the number of trees 

 in a specified diameter class that die in a growth period, and it was modeled as a nonlinear 

 function of number of trees and basal area in the diameter class, and total number of trees 

 and basal area in the stand. Equations were also developed for predicting rough cordwood 

 volume and board-foot volume. Cuttings were introduced by reducing the number of trees in the 

 appropriate diameter class at the end of the growth period. 



As a demonstration of the application of the uneven-aged analytical decisionmaking tools 

 they had developed, Adams and Ek (1974) used a modified version of Ek's (1974) model. 



The difference equation approach was used in this study for two reasons. First, the 

 development of difference equations appeared to be the most logical use of the periodic data 

 available. Because the data source and the equation type are both periodic, it is not necessary 

 to make numerical approximations in order to develop and/or apply the equations. Most models 

 using differential equations do require numerical approximation methods to estimate parameters 

 and/or to solve them. 



Second, the use of difference equations avoids potential problems of inappropriately 

 applying the equations to growth periods different from that used in equation development. 

 Because of the continuity of differential equations, it is possible to predict growth for any 

 period. The differential equations are usually, however, approximations of the true continuous 

 growth process (not only can the parameter estimates be approximations, but the model forms 

 themselves are often approximations). As such, they should not be applied to growth periods 

 much different from those used in equation development or else erroneous results can occur. On 

 the other hand, in normal use difference equations can only be used to predict growth for 

 periods equal to those used to develop the equations. Multiple periodic growth (or fractional 

 periodic growth, if the user wishes to chance extending the equations in that fashion) is 

 derived from these periodic growth rates. 



STUDY DATA BASE 

 Plot Characteristics 



The data used in this study were from the Fort Valley Experimental Forest, located a few 

 miles northwest of Flagstaff, Arizona. The uneven-aged data consist of long-term remeasure- 

 ments on four 80-acre plots. Each plot has been divided into 32 or 34 subplots ranging in size 

 from 0.5 to 3.1 acres, but most of these subplots average near 2.5 acres in size. The measure- 

 ment of the plots started either in 1920 or 1925, and remeasurements have occurred at 5-year 

 intervals until the 1960 's when the interval was lengthened to 10 years. The two attributes 

 recorded on every tree above the lower diameter limit were diameter at breast height (d.b.h.) 

 and tree condition (a classification as to presence of a damaging agent if the tree is alive, 

 or to the cause of mortality if the tree has died within the previous growth period) . Age 

 class and age-vigor class of each tree were measured every fourth growth period. Age class 

 categorizes a tree as being blackjack or yellow pine, and age-vigor uses age and size infor- 

 mation, in combination with the previous 20-year average diameter growth rate, to help identify 

 the tree's growth potential. The lower diameter limit was first set at 3.6 inches (the lower 

 bound for the 4-inch diameter class). It was then changed to 7.6 inches in 1940, and recently 

 to 6.0 inches. A brief history and description of each uneven-aged plot is in table 1. 



2 



