## 3 Mathematical Modeling of Tree-like Fractals
For example, the sequence [\(\alpha_{1}=0\), \(\alpha_{2}\), \(\alpha_{3}\)] together with [\(l_{1}\), \(l_{2}\), \(l_{3}\)] computes the node \(P_{3,1}\), whereas the sequence {\(\alpha_{1}=0\), \(-\alpha_{2}\), \(\alpha_{3}\)} with [\(l_{1}\), \(l_{2}\), \(l_{3}\)] computes the node \(P_{3,3}\)._  
**Property 4**. : _Each parent node \(P_{i,j}\) i > 1, has a set of direct children (connected through a single branch) \(C_{m,n}\) given by:_  
\[C_{m,n}=\{P_{m,n}|m=i+1,\;n=2j-1\ldots 2j\}. \tag{7}\]  
_To obtain the general set \(C_{m,n}\) that defines all children of a node \(P_{i,j}\), the relations of \(m\) and \(n\) (Equation (7)) can be written recursively:_  
\[m=(\ldots(((i+1)+1)+1)\ldots+1), \tag{8}\]  
\[n=(2\ldots(2(2j-1)-1)\ldots-1)\ldots(\ldots(2(2(j)))\ldots),\]  
_which has a recurring relationship:_  
\[m=i+c, \tag{9}\]  
\[n=2^{c}j-2^{c-1}-1\ldots 2^{c}j,\]_where \(c\) is the difference in the nodes' depths. The general set \(C_{m,n}\) of children nodes of a node \(P_{i,j}\) is:_  
\[C_{m,n}=\Big{\{}P_{m,n}\Big{|}m=i+c,\ n=2^{c}j-2^{c-1}-1\ldots 2^{c}j\Big{\}}. \tag{10}\]  
_Finally, a range \(r\) at each depth \(i\) from the total depth \(d\) is defined. The range \(r\) does not have a physical meaning, but it is a useful measure._  
\[r=2^{(d-i)}. \tag{11}\]  
**Property 5**. : _A recursive model can be defined for the tree-like fractal:_  
\[ \begin{split} P_{i}^{\prime}=P_{i-1}^{\prime}\cdot R_{Z}(\alpha_ {i})\cdot T_{Y}(l_{i}),\\ P_{0}^{\prime}=ORG,\ i>0.\end{split}  \tag{12}\]  
\(P_{i}^{\prime}\) _represents nodes in one distinct (without bifurcation) path. Figure 4 highlights an example of a distinct path (from ORG to \(P_{3,2}\)). It is assumed that for any node in the fractal, Equation (12) can be recursively applied._  
**Property 6**. : _Each branch of the fractal is defined by two points: \(P_{i}\) (the initial point of the branch) and \(P_{f}\) (the final point of the branch). A vector can be defined:_  
\[V_{i}=\frac{\Big{(}P_{f}-P_{i}\Big{)}}{\operatorname{norm}\Big{(}P_{f}-P_{i} \Big{)}}, \tag{13}\]  
_representing the direction of the branch, having the length:_  
\[H_{i}=\operatorname{norm}\Big{(}P_{f}-P_{i}\Big{)}. \tag{14}\]  
_Furthermore, as each node is an intersection of three branches (the root and the leaves are not considered), three vectors can be defined:_  
\[ \begin{split} V_{1}=\frac{(p_{1}-P_{1})}{\operatorname{norm}(P_{ 1}-P_{1})},\ V_{2}=\frac{(p_{2}-p_{2})}{\operatorname{norm}(p_{2}-p_{2})},\ V_{3}=\frac{(p_{3}-p_{3})}{ \operatorname{norm}(P_{3}-p_{3})},\\ P_{f1}=P_{2},\ P_{f2}=P_{3},\end{split}  \tag{15}\]  
Figure 4: Schematical representation of a tree-like fractal. Green highlighted path from origin to a point in a fractal tree. _where \(V_{1}\) represents the vector of the parent branch and \(V_{2}\) and \(V_{3}\) represent the vectors of the children branches (\(V_{2}\) points out of the node whereas \(V_{3}\) points towards the node). Using Equation (15), three bisector vectors can be defined (it is assumed that any pair from \(V_{1}\), \(V_{2}\), \(V_{3}\) does not have the same orientation):_  
\[N_{1}=\frac{(V_{1}+V_{2})}{\operatorname{norm}(V_{1}+V_{2})},\ N_{2}=\frac{(V_ {1}+V_{3})}{\operatorname{norm}(V_{1}+V_{3})},\ N_{3}=\frac{(V_{2}+V_{3})}{ \operatorname{norm}(V_{2}+V_{3})}. \tag{16}\]  
**Property 7**. : _With respect to a global coordinate system \(OXYZ\), two angles can be defined for every branch:_  
\[ \begin{split}\psi=\operatorname{atan2}(Y_{V},X_{V}),\\ \varphi=\operatorname{atan2}\Bigl{(}\sqrt{Y_{V}^{2}+X_{V}^{2}},Z _{V}\Bigr{)},\\ V[X_{X},Y_{V},Z_{V}]=\frac{P_{f}-P_{i}}{\operatorname{norm} \bigl{(}P_{f}-P_{i}\bigr{)}},\end{split}  \tag{17}\]  
_where \(P_{i}\) and \(P_{f}\) are the initial and final points of a branch, respectively._  
**Remark 1**. : _The mathematical model can be further generalized by not imposing symmetry. The rotation matrices may not necessarily be evaluated for \(\pm\)\(a_{i}\) but instead with values \(+w_{1}a_{i}\), \(-w_{2}a_{i}\), \(w_{1}\), \(w_{2}\)\(\in\) [0, 1] \(\subset\)\(\mathbb{R}\).