Let $L \subseteq \Sigma^*$ be a regular language. Let $\Sigma' = \Sigma_0 \cup \Sigma_2$ where $\Sigma_0 =\Sigma$ and $\Sigma_2=\{*\}$.
We define $T_L=\{t \in t_{\Sigma'} \mid \text{The leafs from t are from a word in $L$}\}$ as a tree language.
The task is to construct a DTA for $T_L$.
A DTA A over $\Sigma = \Sigma_0 \cup \cdots \cup \Sigma_m $ is a four tuple with $A= (Q,\Sigma, \delta, F)$ where Q is a finite state set, F final state set, and $\delta : \bigcup _{i=0}^m(Q^i \times \Sigma_i)\rightarrow Q$ and $Q^0 \times \Sigma_0 = \Sigma_0$.
My idea was to set $\Sigma _0 \subseteq L \subseteq \Sigma^*$. But how can the transition function check whether it is a word in L?