  In this paper we study pairs of polynomials with a given factorization
pattern and such that the degree of their difference attains its minimum. We
call such pairs of polynomials Davenport--Zannier pairs, or DZ-pairs for short.
The paper is devoted to the study of DZ-pairs with rational coefficients.
  In our earlier paper, in the framework of the theory of dessins d'enfants, we
established a correspondence between DZ-pairs and weighted bicolored plane
trees. These are bicolored plane trees whose edges are endowed with positive
integral weights. When such a tree is uniquely determined by the set of black
and white degrees of its vertices, it is called unitree, and the corresponding
DZ-pair is defined over $\mathbb Q$. In our earlier paper, we classified all
unitrees. In this paper, we compute all the corresponding polynomials.
  In the final part of the paper we present some additional material concerning
the Galois theory of DZ-pairs and weighted trees.
