Smooth Fractal Trees: Analytic Generators and Discrete Equivalence

TL;DR

This paper presents a novel smooth, analytic framework for constructing fractal trees that unifies discrete models with continuous geometric representations, enabling precise analysis of their structure and limits.

Contribution

It introduces a generator-based method for creating smooth fractal trees that can replicate any discrete tree model and analyze their asymptotic geometry.

Findings

Any discrete tree model can be represented by an analytic generator tree.

The accumulation set of branch endpoints matches the discrete attractor.

Fractality depends on recursive branching, not local smoothness.

Abstract

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.