Many divide-and-conquer algorithms employ the fact that the vertex set of a graph of bounded treewidth can be separated in two roughly balanced subsets by removing a small subset of vertices, referred to as a separator. In this paper we prove a trade-off between the size of the separator and the sharpness with which we can fix the size of the two sides of the partition. Our result appears to be a handy and powerful tool for the design of exact and parameterized algorithms for NP-hard problems. We illustrate that by presenting two applications. Our first application is a O(2 n+o(n))-time algorithm for the Degree Constrained Spanning Tree problem: find a spanning tree of a graph with the maximum number of nodes satisfying given degree constraints. This problem generalizes some well-studied problems, among them Hamiltonian Path, Full Degree Spanning Tree, Bounded Degree Spanning Tree, and Maximum Internal Spanning Tree. The second application is a parameterized algorithm with running time O(16 k+o(k)+n O(1)) for the k-Internal Out-Branching problem: here the goal is to compute an out-branching of a digraph with at least k internal nodes. This is a significant improvement over the best previously known parameterized algorithm for the problem by Cohen et al. (J. Comput. Syst. Sci. 76:650–662, 2010), running in time O(49.4k +n O(1)).