Construction of a Deterministic Tree Automaton (DTA)
Your model is a bottom-up tree automaton. Let $P$ be the set of states of the finite state automaton for regular $L$. Now a state of the tree  automaton at a vertex $v$ of the tree contains all $P \times  P$ pairs $(p,q)$ such that there is a path from $p$ to $q$ on the string specified by the leaves under $v$. This can be deterministically evaluated in a bottom-up fashion.

More precisely, with every vertex $v$ we associate a binary relation $R(v)$ over $P$, thus $R(v) \subseteq P\times P$. Thus $(p,q)\in R(v)$ iff there is a path from $p$ to $q$ on the leaves under $v$.

For a leaf $v$ labelled $a\in \Sigma$ this means that $R(v) = \{(p,q)\mid (p,a,q) \in \delta\}$, where $\delta$ is the transition relation of the automaton for $L$.

If $u$ is a binary vertex (labelled by $*$) with children $v,w$, then the relation $R(u)$ is the compostition of $R(v)$ and $R(w)$.

We accept at the root $r$ if $R(r)$ contains an accepting pair $(p,q)$ where $p$ is the initial state and $q$ is final.