Thek-Delaunay tree extends the Delaunay tree introduced in [1], and [2]. It is a hierarchical data structure that allows the semidynamic construction of the higher-order Voronoi diagrams of a finite set ofn points in any dimension. In this paper we prove that a randomized construction of thek-Delaunay tree, and thus of all the orderlek Voronoi diagrams, can be done in O(n logn+k 3n) expected time and O(k2n) expected storage in the plane, which is asymptotically optimal for fixedk. Our algorithm extends tod-dimensional space with expected time complexityO(k lceil(d+1)/2rceil+1 n lfloor(d+1)/2rfloor) and space complexityO(k lceil(d+1)/2rceil n lfloor(d+1)/2rfloor). The algorithm is simple and experimental results are given.