  For any elements b,c of a number field K, let G(b,c) denote the backwards
orbit of b under the map f_c: C-->C given by f_c(x)=x^2+c. We prove an upper
bound on the number of elements of G(b,c) whose degree over K is at most some
constant B. This bound depends only on b, [K:Q], and B, and is valid for all b
outside an explicit finite set. We also show that, for any N>3 and any b in K
outside a finite set, there are only finitely many pairs of complex numbers
(y,c) for which [K(y,c):K]<2^(N-3) and the value of the N-th iterate of f_c(x)
at x=y is b. Moreover, the bound 2^(N-3) in this result is optimal.
