has $l$ children ${\pmb{A}}_{1}(\widetilde{{\pmb{x}}})$, ${\pmb{A}}_{2}(\widetilde{{\pmb{x}}}),\cdots,{\pmb{A}}_{l}(\widetilde{{\pmb{x}}})$, and there is a directed edge from $\widetilde{{\pmb{x}}}$ to ${\pmb{A}}_{i}(\widetilde{{\pmb{x}}})$, labeled by $i$, for $1\leq i\leq l$.

Now we consider the paths in the execution tree. Denote by $\Sigma=\{1,2,...,l\}$ the set of indices of the transformations ${\pmb{A}}_{i}$ ($1\leq i\leq l$). For any string $\sigma=a_1 a_2\cdots a_s\in\Sigma^*$, we write $|\sigma|=s$ for its length and $\operatorname{Pre}(\sigma)=\{a_1\cdots a_i\mid i=0,1,\cdots,s-1\}$ for the set of its proper prefixes. The concatenation of two strings $\sigma$ and $\tau=b_1\cdots b_t$ is written as $\sigma\tau=a_1\cdots a_sb_1\cdots b_t$. For simplicity, we put ${\pmb{A}}_\sigma={\pmb{A}}_{a_s}\circ\cdots\circ {\pmb{A}}_{a_2}\circ {\pmb{A}}_{a_1}$ and ${\pmb{A}}_\epsilon=\mathbf{id}$ (i.e ${\pmb{A}}_\epsilon({\pmb{x}})\equiv{\pmb{x}}$) for the empty string $\epsilon$. Then ${\pmb{A}}_{\sigma\tau}={\pmb{A}}_{\tau}\circ{\pmb{A}}_{\sigma}$. Similarly, for an infinite sequence $\sigma=a_1 a_2\cdots\in\Sigma^\omega$ we define $|\sigma|=\infty$ and $\operatorname{Pre}(\sigma)=\{a_1\cdots a_i\mid i=0,1,\cdots\}$. And the concatenation of a string $\tau=b_1\cdots b_t$ and $\sigma$ is defined as $\tau\sigma=b_1\cdots b_ta_1a_2\cdots\in\Sigma^\omega$.

Then any finite or infinite path from the root in the execution tree can be identified by a finite or infinite string over $\Sigma$. Specifically, for any $\sigma=a_1a_2\cdots\in\Sigma^*\cup\Sigma^\omega$, the corresponding execution path is as follows: $${\pmb{x}}\xrightarrow{{\pmb{A}}_{a_1}}{\pmb{A}}_{a_1}({\pmb{x}})\xrightarrow{{\pmb{A}}_{a_2}}{\pmb{A}}_{a_2}({\pmb{A}}_{a_1}({\pmb{x}}))\xrightarrow{{\pmb{A}}_{a_3}}\cdots.$$ Moreover, any node in the execution tree is of the form ${\pmb{A}}_\sigma({\pmb{x}})$, where $\sigma\in\Sigma^*$ represents the history of the execution. Its ancestor nodes are ${\pmb{A}}_\tau({\pmb{x}})$ ($\tau\in\operatorname{Pre}(\sigma)$). According to the definition of execution tree, we 