## 1 Introduction  
In China, with the advancement of urbanization and the development of rural revitalization strategies, traditional settlements are confronted with the conflicting challenges of preservation and renewal. The urgency to protect these traditional settlements during the renewal process is paramount. The value of preserving traditional settlements lies in both the high-value individual buildings and courtyards within them, as well as their overall spatial configuration [1, 2]. While individual structures tend to receive greater attention during renewal efforts and preservation work, the latter--namely, the overall spatial morphology of these settlements--faces a considerable risk of being compromised amidst rapid urbanization and insufficient awareness of conservation [3]. The spatial morphology of traditional settlements, in comparison to the majority of modern villages, is capable of preserving more authentic spatial information. Specifically, mountainous traditional settlements, constrained by transportation and topographical factors, experience lower levels of external interference than their plain counterparts. As a result, they retain a richer repository of embedded information, serving as critical samples and references for the study of spatial morphology conservation in traditional settlements [4]. To protect spatial morphology, it is essential to conduct both qualitative and quantitative analyses. This study aims to explore the spatial morphology of mountainous settlements from a quantitative analysis perspective, providing reference data for settlement morphology preservation. While significant foundational research has been conducted by scholars both domestically and internationally regarding qualitative analysis methods for spatial morphology, quantitative analysis primarily focuses on morphological indices and factor relationship analyses [5]. The initial exploration of spatial morphology indices relied on Euclidean geometry, which is manifested in fundamental attributes such as distance and area [5; 6]. However, spatial morphology cannot be fully encapsulated by these basic properties. As complex systems, structures such as snowflakes, branching trees, and mountain ranges defy comprehensive description using suitable indices derived from Euclidean geometry. With ongoing scholarly research, fractal geometry has been increasingly demonstrated to effectively address such complex problems. By introducing fractal dimension as a mathematical tool, researchers have been able to quantify the characteristics of complex systems that are difficult to describe using traditional methods, thereby providing a scientific basis and practical means for further investigation [7]. According to existing research, fractal theory has been widely applied in urban systems. In 1985, [PERSON] introduced the concept of fractal cities and utilized fractal theory to simulate urban land use and activity patterns [8]. In 1989, [PERSON] employed four different methods to calculate the fractals of cartographic lines in Swinton, UK, and examined the functional relationships between fractal dimensions and various factors [9]. Additionally, [PERSON] investigated the expansion patterns of construction land in both urban and non-urban areas of Wentai region in Eastern China along with their related factors [10]. It is evident that the application of fractal dimensions within urban systems is extensive, encompassing a range of aspects from initial analyses of overall urban morphology to exploring influencing factors and development models. In the realm of rural and settlement studies, scholars have introduced fractal geometry theory into the analysis of rural spatial morphology. [PERSON] employed fractal geometry to investigate three elements: traditional settlement boundaries, buildings, and building complexes, thereby establishing a quantitative framework [11]. [PERSON] conducted research on the morphological differences between urban and rural settlements in mountainous and plain regions [12].