How to constrain a finite automaton (NFA and DFA) to a tree?
I think the easiest way of enforcing tree shape is the set of conditions


$q_0$ is not in the image of $\delta$, 
$\delta$ is injective, and
$M$ is connected (to avoid isolated cycles). Note that this one is global, not local, which may be unavoidable.


Then we can prove (by induction) that for any state $q$ there is a unique path from $q_0$ to $q$.