This paper introduces two efficient algorithms that compute the Contour Tree of a 3D scalar field F and its augmented version with the Betti numbers of each isosurface. The Contour Tree is a fundamental data structure in scientific visualization that is used to pre-process the domain mesh to allow optimal computation of isosurfaces with minimal storage overhead. The Contour Tree can be also used to build user interfaces reporting the complete topological characterization of a scalar field, as shown in Figure 1. In the first part of the paper we present a new scheme that augments the Contour Tree with the Betti numbers of each isocontour in linear time. We show how to extend the scheme introduced in 3 with the Betti number computation without increasing its complexity. Thus we improve on the time complexity from our previous approach 8 from 0(m log m) to 0(n log n+m), where m is the number of tetrahedra and n is the number of vertices in the domain of F. In the second part of the paper we introduce a new divide and conquer algorithm that computes the Augmented Contour Tree for scalar fields defined on rectilinear grids. The central part of the …