Applying pareto frontier theory and ball tree algorithms to optimize growth boundaries for sustainable mountain cities

The terrain complexity of mountain cities often limits the rational layout and agglomeration distribution of newly added urban land. To this end, this paper proposes an innovative urban growth boundary optimization model -- Pareto Frontier theory and Ball-Tree algorithm optimized model (SP-BT). By treating newly added urban land as agents, the model dynamically optimizes the land patch locations based on the two goals of urban landscape compactness degree and suitability for urban construction. The SP-BT model is unique in that its optimal search and update process always unfolds along the Pareto front, achieving a balance between urban land agglomeration and suitability. In addition, compared with Suitability Optimum Evaluation Model (SOE), Artificial Neural Network in Cellular Automata (ANN-CA), and Ant Colony Optimization Model (ACO), the model has simplified parameter setting, fast evolution speed, flexible output results, and excellent defragmentation effect, especially suitable for mountain cities under complex terrain conditions. In the empirical study taking Bazhong city, China as an example: (1) SP-BT model significantly increased the mean size of urban landscape patch from 8.86 to 22.74-95.88, and the decrease of urban total suitability was controlled at 3.14%. (2) The SP-BT model excels in handling urban planning in mountainous cities, particularly in reducing landscape fragmentation, making it suitable for practical urban planning in mountainous environments. In general, the model proposed in this paper provides an efficient and flexible solution to the problem of new land use optimization under the complex terrain conditions of mountainous cities, which has important practical application value and can provide planners with more scientific and practical decision support tools.