174 tion of city records, historic data and imagery, local tree groups, and coring and cross-dating tree rings. Equations to predict DBH from age, and tree height, crown height, crown diameter, and leaf area from DBH were developed using least-squares linear and non-linear regressions to determine best fit. Over 1,800 equations are currently used within i-Tree Streets to predict growth and esti- mate the benefits and costs associated with street tree populations, including those published for California’s San Joaquin Valley and Southern Coastal regions (Peper et al. 2001a; Peper et al. 2001b). Testing the applicability of i-Tree Streets outside of the United States, Soares and others (2011) compared growth of urban spe- cies and genera from Lisbon, Portugal, to growth models used in i-Tree Streets. Finding similar growth for U.S. species from Med- iterranean climates and lacking age-based growth curves for most species in Lisbon, the authors used growth curves for U.S. species. Lindenmayer-Systems Lindenmayer-Systems (L-Systems) are a mathematically-based theory of biological development used to model tree growth. L-systems describes the complex growth of trees using produc- tion rules that govern branching and growth at successive time steps (Prusinkewicz and Lindenmayer 1990). Brasch and others (2007) created empirical tree growth models with L-Systems so that modeled crown dimensions conformed to measured di- mensions. Their approach based global characteristics such as tree height, crown diameter, and shape on empirically derived growth equations for nine deciduous species in Modesto, CA, (Peper et al. 2001a). These species’ specific global characteris- tics informed and constrained models of tree growth that were driven by L-Systems’ local production rules (Rudnick et al. 2007). Using 20 time steps per year for 40 years, branch pro- duction rules were applied to all branch segments based on the biological response of each organ to competition for light and space. Computer visualizations produced relatively accurate and life-like growth for each species. This work is important because it was the first to combine urban tree growth equations with L-Systems and computer animation software. It created realistic visualizations of long-term tree growth in a matter of seconds. Other Empirical Models Stoffberg and others (2008) developed tree height and crown size equations for three street tree species in Tshwane, South Africa based upon the same analytical methods used for i-Tree Streets tree growth predictions. Similarly, Semen- zato and others (2011) included the logarithmic regression model proposed by Peper and others (2001a) in developing growth predictions for five important Italian urban species. Process-Based and Hybrid Models These types of models explicitly describe how the uptake and assimilation of carbon and other resources (e.g., nutrients, light, water) translate into morphological growth. Process- based models need many parameters to characterize a species, but modeling of processes need not be parameterized for each species once they are adequately described. Process model- ing can define key growth parameters to help identify traits to measure. An advantage of process-based models is their abil- ity to model the effects of different doses of stressors, such as ©2012 International Society of Arboriculture McPherson and Peper: Urban Tree Growth Modeling with climate change, to identify physiological response thresh- olds. Disadvantages of process-based models are their com- plexity, which put a high demand on computer resources and cause difficulty quantifying uncertainty in model predictions. L-PEACH The L-PEACH functional-structural model was developed to better understand the effects of management decisions on the development, growth, and fruit yield of peach trees (Allen et al. 2005). It treats the tree as a network of semi-autonomous components that interact with each other and the environment. An electrical circuit analogy is used to compute the flow of carbohydrates among components. The most recent enhance- ment added a xylem circuit to simulate water uptake over a season and the effect of different irrigation regimes on growth and yield (Da Silva et al. 2011). A comparison of model outputs and measured water use indicated that the model successfully coupled water transport with growth. L-PEACH’s 3D visualiza- tion makes it an educational tool for growers and students. Fu- ture development will improve light and root system modeling. Hybrid Models Valentine and Mäkelä (2005) suggest broaching the complexi- ties of process-based modeling in modeling forest stands by developing a process-based model fitted and applied in an empirical mode. Their carbon-based model of tree growth in- corporates minimal levels of structure and function, and uses commonly inventoried state variables—tree height, crown height, and stem cross sectional area. This is a straightforward and efficient attempt to address the frustration confronting all growth modelers—the absence of a robust model that melds fundamental biological knowledge of tree growth with a func- tional balance approach to modeling. Forest modelers face the frustration of having multiple models for the same species growing in the same habitat. Valentine and Mäkelä’s work at- tempts to resolve this issue. Their methods may be applicable in urban settings where biotic and abiotic stressors are necessary to explain differences in tree growth, longevity, and mortality. Other Approaches Efficient methods for collecting field data for modeling, moni- toring, and evaluating growth influences remain problematic. Municipalities often cannot afford to conduct inventories or col- lect data beyond what is required for the basic management of their tree populations. In addition to presenting growth research, the following studies took different approaches in efforts to re- solve the data collection issue. Jutras (2008) conducted a com- prehensive study on the influence of multiple biotic and abiotic factors on the growth of seven street tree species in Montreal, Canada. Using artificial intelligence and multivariate statistics, he accomplished three objectives. First, he found a combina- tion of 11 biotic descriptors that illustrate all tree physiologi- cal stages. Next, he applied contingency analysis to determine links between abiotic variables, including urban zone type, surficial deposits, solar irradiation level, street width, distance from tree to curb, and tree growth. Last, he sought to optimize tree inventory procedures using aerial Light Detection and Ranging (LIDAR) combined with other less field work inten- sive variables. Ultimately, the efficiently collected data and the
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