Terrain trees are a new in-core family of spatial indexes for the representation and analysis of Triangulated Irregular Networks (TINs). Terrain trees combine a minimal encoding of the connectivity of the underlying triangle mesh with a hierarchical spatial index, implicitly representing the topological relations among vertices, edges and triangles. Topological relations are extracted locally within each leaf block of the hierarchal index at runtime, based on specific application needs. We have developed a tool based on Terrain trees for terrain analysis, which includes state-of-the-art estimators for slope and curvature, and for the extraction of critical points, as well as algorithms for topology-based terrain segmentation and multifield terrain analysis. By working on TINs generated from very large LiDAR (Light, Detection and Ranging) data sets, we demonstrate the effectiveness and scalability of the Terrain trees against a state-of-the-art compact data structures.