Let φ : D → R be a continuous scalar field where D is a rectangular region in R d. More generally, D is homeomorphic to a closed ball in Rd. A connected

component of a level set φ−1(σ) is a maximally connected subset of φ−1(σ), i.e., a connected subset of φ−1(σ) that is not contained in any other connected subset of φ−1(σ). As σ changes, the connected components of φ−1(σ) change. New components appear, existing components disappear, multiple components join together to form one component, or a single component splits apart into multiple components. A contour tree is a graph that represents these components and their changes.