There are some essential differences between the criticality tree and the contour tree of [4], [5], [6] (see figure 3). The contour tree traces the evolution of individual contours (boundary manifolds), whereas the criticality tree traces the evolution of the objects they bound. The contour trees of [4], [6] do not record changes in genus, only the creation, destruction, splitting, or merging of components. Another difference is that leaf nodes in the contour tree are maxima and minima, where manifolds are created or vanish, while in our criticality tree leaf nodes are all maxima, since decreasing the threshold can create an object, but cannot cause an object to vanish or split. In the criticality tree a node with multiple children only occurs when two distinct objects merge, not when two boundary manifolds of the same object merge as in the contour tree. In the criticality tree a merging or splitting of the boundary manifolds of a single object will yield a parent with a single child. Finally, our zone components contain a more refined partition of the nodes along a superarc of the augmented contour tree (called a join set in [4]) as well as other nodes not included on a superarc. This because we include genus changing saddle values, and, a zone component contains only the portion of space swept by a contour for the range of thresholds for which which the entire boundary of the object it bounds remains topologically invariant. Also we include all boundary points of the zone (see section 5 below).