We introduce a framework for constructing fractal trees via analytic generator fields, replacing discrete affine transformations and symbolic rewriting rules by the integration of smooth vector fields in an internal state space. In this setting, geometric curves are obtained as projections of generator trajectories, and branching is implemented as a primitive operation through exact inheritance of generator state. At every finite depth, the resulting structure is a finite union of analytic curve segments that is smooth across branch events. Two structural results relate this generator-driven construction to classical discrete models of tree-based fractals. First, a combinatorial universality theorem shows that any discrete tree specification, including those arising from iterated function systems and L-systems, can be compiled into an analytic generator tree whose induced discrete scaffold is isomorphic at every finite depth. Second, under standard contractive assumptions, a canopy set equivalence theorem establishes that the accumulation set of analytic branch endpoints coincides with the attractor of the corresponding discrete construction. These results separate local geometric regularity from global fractal complexity, showing that fractality is determined by recursive branching and scaling rather than by local non-smoothness. The framework provides a smooth representation of tree-based fractals that preserves both their finite combinatorial structure and their asymptotic limit geometry.

翻译：我们提出了一种通过解析生成器场构建分形树的框架，用内部状态空间中光滑向量场的积分取代了离散仿射变换和符号重写规则。在此设定下，几何曲线通过生成器轨迹的投影获得，而分支则通过生成器状态的精确继承作为基本操作实现。在任意有限深度下，所得结构均为解析曲线段的有限并集，且在分支事件处保持光滑性。两项结构结果将此生成器驱动的构造与经典的树状分形离散模型联系起来。首先，组合普适性定理表明：任何离散树规范（包括迭代函数系统和L-系统产生的规范）均可编译为解析生成器树，其诱导的离散支架在所有有限深度上保持同构。其次，在标准压缩假设下，冠层集等价定理证明解析分支端点的累积集与对应离散构造的吸引子完全一致。这些结果将局部几何正则性与全局分形复杂性分离，表明分形性由递归分支和缩放决定，而非由局部非光滑性决定。该框架为树状分形提供了光滑表示，同时保留了其有限组合结构与渐近极限几何。