Irreducible polynomials over finite fields produced by composition of quadratics

For a set S of quadratic polynomials over a finite field, let C be the (infinite) set of arbitrary compositions of elements in S. In this paper we show that there are examples with arbitrarily large S such that every polynomial in C is irreducible. As a second result, when #S>1, we give an algorithm to determine whether all the elements in C are irreducible, using only O(#S(logq)3q1/2) operations.