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Sec. 13.3 Spatial Data Structures 447CADBEFFigure 13.14 Searching in the k-d treeof Figure 13.11. (a) The k-d tree decompositionfor a 128 × 128-unit region containing seven data points. (b) The k-d treefor the region of (a).private void rshelp(KDNode rt, int[] point,int radius, int lev) {if (rt == null) return;int[] rtkey = rt.key();if (InCircle(point, radius, rtkey))System.out.println(rt.element());if (rtkey[lev] > (point[lev] - radius))rshelp(rt.left(), point, radius, (lev+1)%D);if (rtkey[lev] < (point[lev] + radius))rshelp(rt.right(), point, radius, (lev+1)%D);}Figure 13.15 The k-d tree region search method.Figure 13.15 shows an implementation for the region search method. Whena node is visited, function InCircle is used to check the Euclidean distancebetween the node’s record and the query point. It is not enough to simply checkthat the differences between the x- and y-coordinates are each less than the querydistances because the the record could still be outside the search circle, as illustratedby Figure 13.13.13.3.2 The PR quadtreeIn the Point-Region Quadtree (hereafter referred to as the PR quadtree) each nodeeither has exactly four children or is a leaf. That is, the PR quadtree is a full fourwaybranching (4-ary) tree in shape. The PR quadtree represents a collection ofdata points in two dimensions by decomposing the region containing the data pointsinto four equal quadrants, subquadrants, and so on, until no leaf node contains morethan a single point. In other words, if a region contains zero or one data points, then