A mathematical model has been developed to describe the growth of trees in stands of Douglas fir (Pseudotsuga menziesii (Mirb.) Franco) from age ten to age 100 years. An initial square pattern of spacing was assumed. At age ten years the trees were assumed to be open-grown, that is, growing in diameter at breast height at a maximum rate. A regression of d.b.h. on age was obtained from eighteen open-grown, Douglas fir trees measured on the Saanich Peninsula, Vancouver Island. The relationship derived from these data agreed with further data collected elsewhere in the coastal regions of British Columbia and Washington and in the interior of British Columbia. The d.b.h. growth of individual trees was predicted by five-year periods. Relationships between crown width and d.b.h. were calculated from data on 426 open-grown, Douglas fir trees. There was a close correlation between crown width and root spread for open-grown trees. A multiple regression equation was obtained for height of 869 trees on d.b.h. and basal area per acre. All regression equations calculated for use in the model, were highly significant statistically. The model is initiated with a matrix of 15 x 15 trees (or tree "locations”). The initial d.b.h. of each tree is specified and, from the crown width/d.b.h. regressions, the crown width of each tree is calculated. As long as the tree remains free of competition, this calculated crown width is reduced by 40 per cent by the reduction factor "REDFAC", to give the "competitive" crown width. This was because it was found that, in young Douglas fir plantations, there could be considerable overlapping of the crowns before d.b.h. growth was reduced. As soon as competition sets in the original 40 per cent reduction is systematically reduced. The proportion of the circumference of each tree that is occupied by the crowns of surrounding competitors is then calculated. This proportion indicates the amount of competition to which the tree is being subjected and varies between zero, if the tree is open-grown, and one or more, if the tree is completely enclosed by the surrounding competitors. If the reduction is sufficiently great, continued survival of the tree is considered unlikely, and the tree is assumed to have died. The periodic d.b.h. growth of the surviving trees is calculated at five-year intervals to age 100 years. All calculations are performed using am I.B.M. 7090 electronic computer. A summary of the structure of the stand can be printed out at the end of each five-year period if required. Height growth can be described by modifying the stand model by including an appropriate regression equation. Similarly, volume growth can be estimated by modifying the basic stand model. The mathematical model developed here satisfactorily describes the growth of Douglas fir stands on an individual tree basis, over a wide range of site conditions, stand densities, amounts and distributions of mortality and thinning regimes. Field data cannot be secured to evaluate the accuracy of all the tests made. However, there are no gross errors in absolute values and results are accurate proportionately. The model described here can aid the forester in managing Douglas fir stands in the Pacific Northwest. By simulating the growth of his stands from age ten to age 100 years in a few minutes he can study questions that would otherwise require several human generations to evaluate. / Forestry, Faculty of / Graduate

In this thesis the five equation system of growth and yield equations originally developed by Clutter (1963) is examined. The system is redeveloped algebraically to form a truly algebraically compatible system. Three methods of estimating the coefficients were examined. In the first method, three of the equations were fitted independently using ordinary least squares; these coefficient estimates were carried through to the other equations. No consideration was given to the relationships that must exist between the equation coefficients in order for the system to be numerically consistent. In the second method the system is first developed algebraically, before any of the coefficients are estimated, resulting in a slightly different system which is truly algebraically compatible. The coefficients were estimated by fitting two of the equations, and using these estimates throughout the rest of the system. The resulting system is both numerically consistent and algebraically compatible. In the final method the relationships between the coefficients that must hold for the system to be compatible were incorporated in the coefficient estimation procedure. Seemingly unrelated regression techniques were used to estimate the coefficients. The three methods resulted in coefficient estimates that were similar, with seemingly unrelated regression producing the most efficient estimators. Prediction ability of the three methods on independent data show no method as being superior, although the seemingly unrelated regression procedure was able to reduce the total system error best. / M.S.

An integrated system of stand models was developed for loblolly pine in which models of different levels of resolution are related by a unified mathematical structure. A “telescoping'' system is presented in which a highly detailed overall model is specified and its components. ''collapsed" around a common set of growth and survival functions to provide structurally compatible models at each successively lower stage of resolution. The most detailed model is a distance dependent individual tree model which simulates the growth and competitive interaction of trees in a stand. Tree basal area and height growth were modeled using a modified Chapman-Richards function in terms of potential growth, current size, relative size, crown ratio, and an index of competition. Potential growth was considered a function of site quality, age, and open-grown size relationships. Tree survival probability was described using a logistic function in terms of age, crown ratio, and competition. The competition or point density index is a function of the size and location of neighbors. Published indices were evaluated and compared on their simple correlation with growth, multiple correlation with growth in the presence of other tree and stand measures, computer execution time, and relationships to stand level density measures. The area potentially available (APA) for each tree was chosen as the most suitable. The APA index is calculated as the area of the polygon constructed from lines which divide the distance between a tree and its neighbors. Mean APA, or average area per tree, is estimated by the inverse of trees per unit area, permitting point density to collapse to stand density, resulting in a distance independent individual tree model. This model was collapsed dimensionally to consider trees grouped in size classes. Tree growth and survival equations are applied to the mean attributes of each size class, resulting in a size class projection model. At the lowest level of resolution, the dimensions of the model are collapsed to one "average" tree. A stand level projection model results from applying the tree growth and survival equations to the stand's average tree attributes. At the stand level, the basal area growth function provides a transformation which, for a number of probability density functions (pdf's), will regenerate the initial pdf family. Considering a normal pdf to describe basal area distributions, a pdf-based size distribution model is presented, in which the projected parameters are expressed in terms or the growth function coefficients. Applications to other pdf families are discussed. Preliminary tree growth and survival equation coefficients were estimated using data from a loblolly pine stand density study in North Louisiana. Structurally compatible models at each level of resolution are detailed. Considerations for numerically consistent estimates from models of different levels of resolution are discussed in terms of model specification, estimation, and implementation. Recommendations for model application and future model development are presented. / Ph. D.

Stem profile models which allow for both taper and form changes (Gray 1956) were constructed and evaluated. Gray defined form to be the basic shape of the tree, e.g. cone or parabolid, and taper to be the rate of narrowing in diameter given a tree form. Ormerod's stem profile model was selected as the basic model since its parameters were readily interpretable in terms of Gray's taper and form definitions. Two stage modeling procedures were used to relate individual tree taper and form parameters to tree and stand characteristics. Two second-stage parameter estimation alternatives were evaluated. Parameter estimates for both techniques, ordinary least squares and random function analysis, were similar. Characteristics used to predict stem form were total tree height, crown ratio, height to the live crown, site index, and tree age. The taper parameter was related to diameter at breast height, crown ratio and site index. Error evaluations suggest that substantial gains in predicting stem diameters were not made using the variable taper and form stem profile models. Two methods were proposed for modeling whole stem inside-bark diameter or cross-sectional area increment. Whole stem increment models were derived from several stem profile models and Presseler's hypothesis on the vertical distribution of cross-sectional area growth. Stem profile models evaluated for constructing compatible increment models were Kozak and others (1969), Ormerod (1973), Goulding and Murray (1976), Max and Burkhart (1976), Cao and others (1980), and Amidon (1984). The increment model based on Presseler's hypothesis was derived as a generalization of the work of Mitchell (1975). Evaluations, with limited increment data, consistently showed that the models based on Presseler's hypothesis predict inside-bark diameter increment with less error than do the profile model compatible increment models. This may be due to the lack of crown information currently used in stem profile models. / Ph. D.

Trees represent important living components in many urban parkland spaces. As living landscape entities, they have the capacity for potentially long life spans. As a result of these longevities, issues concerning tree death or senescence are not often engaged until the end of tree life spans have been reached, or are fast approaching. As organisms with finite life spans, tree senescence must be expected at some future point in time, and due consideration of this inevitable change is imperative within an urban parkland context. An understanding of tree longevity in urban parkland spaces must therefore be considered advantageous to subsequent design, management, and planning decisions enacted upon these landscapes. For appropriate decision-making to take place with regard to urban tree populations, figures reflecting expected tree longevity could purvey estimations of future tree senescence, and assist in providing practical information for all stakeholders of urban landscapes. In addition to this, developed models of parkland spaces supplying visual and spatial analysis of future tree senescence patterns could indicate potential landscape scenarios, and highlight tree populations most at risk of senescence within the near future. The development of models predicting possible future tree senescence patterns required a review of various fields of research in order to establish appropriate models for use, and to assign confidence levels based upon the knowledge of tree growth, longevity, and senescence in predicted landscapes. This thesis examined the subjects of tree longevity and senescence, with a particular focus upon the Adelaide Park Lands region in Adelaide, South Australia. Various tree growth parameters were collected from the field and combined with assigned tree ages to create matrix models that represented expected tree growth trends. Through the incorporation of curves fitted to these matrix models, tree ages could be assigned to tree specimens of unknown age, to determine dates of establishment based upon key growth parameters. Tree longevity figures for each taxon were sourced from a peer reference group survey conducted specifically for this purpose. Through the combination of calculated tree age and predicted tree longevity, senescence patterns for a region of the Adelaide Park Lands were modelled. Interactive structured query-based GIS software was incorporated to display these senescence patterns visually, and to provide interpretations of future landscape scenarios. Results obtained from the peer reference group survey provided a range of valuable figures representing expected tree longevities for 131 taxa from within the Adelaide Park Lands environment. These longevity figures, combined with matrix models and GIS simulations, revealed that considerable populations of established trees within Tuttangga/Park 17 in the Adelaide Park Lands are at a high risk of reaching senescence within the near future. / Thesis (Ph.D.) -- University of Adelaide, School of Architecture, Landscape Architecture, and Urban Design, 2008

Growth responses to climatic variables of red pine (Pinus resinosa Aiton) and Norway spruce (Picea abies L. Karst) were studied at the Morgan Arboretum, near Montreal, in southern Quebec, Canada (45$ sp circ$ 25$ sp prime$ N, 73$ sp circ$ 57$ sp prime$ W; 15.2 m above sea level). The relationships between climatic variables and basal area growth were examined using linear and quadratic models. Current and previous year's climatic variables were tested separately and in combination using multiple regression models. The best models explained 82% and 85% of the total variance of the growth of Norway spruce and red pine, respectively. The growth of both species was more associated with evapotranspiration than precipitation. The growth of Norway spruce was best explained by the current year's annual evapotranspiration (43%), while the growth of red pine was more related to previous year's August evapotranspiration (33%) at our site. / The JABOWA model was used to predict tree growth in hypothetical climates which could result from global climate changes. Based on literature, five treatments were applied: normal, and increases of 1, 3, 5, and 10$ sp circ$C. Comparison between the last (1983-1992) and next (1993-2002) ten years growth showed no significant differences between species when temperature was normal or increased by 1 and 3$ sp circ$C, but significant differences between species were observed when the temperature was increased by 5$ sp circ$C. Both species declined when the temperature was increased by 10$ sp circ$C. We concluded that Norway spruce is more sensitive to increases in atmospheric temperatures than red pine at our site.

A simplified process-based model simulating growth and water use in forest plantations was utilised to predict the growth of Pinus elliottii in South African forest plantations. The model is called 3-PG (Physiological Principles in Predicting Growth) and predicted the growth of trees by simulating physiological processes that determine growth and water use, and the way trees are affected by the physical conditions to which they are subjected, and with which they react. Pinus elliottii growth data recorded in 301 sample stands around South Africa were sourced from forestry companies. A selection procedure reduced the number of stands to 44, where 32 were used to parameterise 3-PG and 12 were reserved for testing the final model parameters. This was accomplished by matching model output to observed data. All stand simulations were initialised at age four years and continued to the maximum age of recorded growth. A provisional set of parameter values provided a good fit to most stands and minor adjustments of the specific leaf area (σ), which was assigned a value of 5 m2.kg-1, were made, bringing about an improved fit. The predictions of mean DBH, Height, and TPH were relatively good, achieving R2 of 0.8036, 0.8975, and 0.661 respectively, while predictions of stem volumes were worse (R2 =0.5922, n=32). The 3-PG model over-predicted DBH in 20 stands, while modelled volume predictions improved substantially in thinned stands (R2 =0.8582, n=14) compared to unthinned stands (R2 =0.3456, n=18). The height predictions were generally good producing an R2 =0.8975. The final set of 3-PG parameter values was then validated against growth data from the 12 independent stands. The predictions of mean DBH, Height, and TPH were relatively good, achieving R2 of 0.8467, 0.7649, and 0.9916 respectively, while predictions of stem volumes were worse (R2 =0.5766, n=12). The results of this study demonstrated the potential for 3-PG to respond to many growth factors and to predict growth and water use by trees with encouraging realism. Patterns of changing leaf area index (L) over time, responses to drought, and annual evaporation patterns all look realistic. Consequently, 3-PG is judged to have potential as a strategic forestry tool. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2011.

No description available.

A stand-level growth and yield model was developed to predict future diameter distributions of thinned stands of mixed Appalachian hardwoods. The model allows prediction by species groups and diameter classes. Stand attributes ( basal area per acre, trees per acre, minimum stand diameter, and arithmetic mean dbh) were projected through time for the whole stand and for individual species groups. Future diameter distributions were obtained using the three-parameter Weibull probability density function and parameter recovery method. The recovery method used employed the first two non-central moments of dbh (arithmetic mean dbh and quadratic mean dbh squared) to generate Weibull parameters. Future diameter distributions were generated for the whole stand and every species group but one; the diameter distribution of the remaining species group was obtained by subtraction from whole stand values. A system of tree volume equations which allow the user t o obtain total tree volume or merchantable volume to any top height or diameter completes the model. Volumes can be calculated by species group and summed to get whole stand volume. / M.S.