The Delaunay is a hierarchical data structure that has been introduced. The Delaunay tree is defined from the Delaunay triangulation and, roughly speaking, represents a triangulation as a hierarchy of balls. It allows us to incrementally construct the Delaunay triangulation of a finite set of n points in any dimension. In this paper, we prove that a randomized construction of the Delaunay tree (and thus, of the Delaunay triangulation) ca be done in 0(n log n) expected time in the plane and in [??] expected time in d-dimensional space. These results are optimal for fixed d. The algorithm is extremely simple and experimental results are given.