Bottom-Up derivatives of tree expressions

In this paper, we extend the notion of (word) derivatives and partial due to (respectively) Brzozowski Antimirov tree using already known inductive formulae quotients. We define a new family extended regular expressions (using negation or intersection operators), show how compute Brzozowski-like automaton; fixed point construction, when it exists, is derivative automaton. Such deterministic automaton can be used solve membership test efficiently: whole structure not necessarily computed, computations performed in parallel. also our (Bottom-Up) derivatives, without computing an