## 1 Introduction  
Additive manufacturing (AM), also called 3D printing, is a revolutionary technology that enables the manufacturing of mechanical components with complex geometry (unfeasible using milling or molding). Many materials are available in AM, ranging from plastic (applied layer by layer) to ceramics or metal alloys [1; 2] (e.g., selective laser melting). Furthermore, AM allows the for the design and manufacture (or even the repair [3]) of components with internal structures for which the aim is to achieve various specific mechanical properties. In [4], the authors detail different lattice structures achieved using AM, highlighting the mechanical properties of these structures. In a comprehensive review article [5], the authors studied the mechanical properties of gradient lattice structures available in the literature. One point, made in [5], is that many lattice structures are inspired by nature (biomimicry). A second point is that lattice structures have the potential to create lightweight and strong mechanical parts. Other authors have proposed modeling and optimization methods for lattice structures [6; 7; 8; 9]. Another class of internal structures for AM studied in the literature contains fractal structures. Fractals (scale-invariant repetitive structures) are also structures inspired by nature. The scientific literature regarding the use of fractals in AM is significantly thinner than that regarding the use of lattices. However, the following are some reports regarding the use and study of fractal structures for AM. In [10], the authors describe a cross-based fractal designed for shock absorption, showing convincing results (for shock absorbing helmet). At the micro level, fractal structures have also been proposed for the generation of porous structures [11]. More recently, Hilbertcurves have been proposed for internal structures in structured materials [12] with promising results in the material mechanical properties. It is worth noting that the authors in [9] also point out a lack in the use of fractals for internal structures in the current literature. Hilbert curve fractals are also studied in [13] for energy dissipation purposes. The authors conclude that the geometry of these structures influences the mechanical properties of the metamaterial. Tree-like fractal structures are also studied in the literature (though only briefly). In [14], the authors used tree-like fractal structures as support structures in AM, showing various application examples (and concluding that this approach impacts cost reduction and the environment). The scientific literature points out that, although tree-like fractal structures can be considered for internal structures in AM parts, their study is in its infancy. Indeed, in [15], the authors point out the remaining potential and the challenges for the future development of AM technologies. More understanding (mechanical behavior, etc.) is required for AM parts with internal structures. In a nutshell, there is a blind spot with respect to the properties and applications of tree-like fractals (as internal structures) for AM, and this work aims to establish a model to generate such internal structures so as to open a window of opportunity to study them. Whether or not tree-like fractals have better mechanical properties (for a given domain of applications) remains the subject of further research. The hypothesis is that tree-like fractals perform better than lattice structures in non-homogenous meta-materials, e.g., 3D-printed bone or parts that must deform in a predictable manner under mechanical stress for safety reasons (e.g., helmets, vehicle components). In [16], the authors present a mathematical model that generates tree-like fractal structures using the language of homogeneous transformation matrices (which have various technical applications, e.g., mechanism analysis [17; 18]). To the authors' best knowledge, there is no other such model in scientific literature.