Understanding Applications of Binary Search in Computing Order Statistics A fundamental idea in computer science, binary searches are essential to many algorithms and data structures. The uses of binary searches in computing order statistics and the use of binary search trees in storing assorted lists will be the main topics of this paper. How do Order Statistics work? An individual ranking inside a collection of numerical or categorical variables is referred to as an order statistic. The order statistics problem, where k is an integer between 1 and n, asks for the kth lowest or greatest element in a set of elements (the total number of elements in the set). The order statistic problem, for instance, arises when determining the median, the 25th percentile, or the seventh largest element in a set. Making Order Statistics Using Binary Searches It is fundamentally a search problem to approach computing order statistics using binary searches. The procedure should provide the kth smallest element kept in the tree given the root of a binary search tree and the number k. To accomplish this, we must first determine which subtree to look in. In order to solve this issue, a new field called N.Size that returns the number of entries in the N subtree must be added. We must add an operation called RecomputeSize to verify that the value of the field N.Size is accurate at all times. The size of a node is recalculated in this operation as the sum of the sizes of its left and right children plus one. The widths of the subtree will vary when the tree is rotated, thus we must recalculate the sizes of the two rotated nodes. Once we have all of the nodes' sizes stored, computing order statistics is rather simple. We first determine the size of the left subtree, R.Left.Size, in order to calculate the kth lowest element in the tree. The root is the 𝑠 + 1st smallest element, and we return R if 𝑘 = 𝑠 + 1, where s is the size of the left subtree. If k s + 1, the left subtree has the kth smallest element, and we recurse on the left child. But if 𝑘 > 𝑠 + 1, the kth lowest element is in the right subtree, and we recurse on the right child while decreasing k by (𝑠 + 1). Binary Search Trees for Assorted List Storage Assorted lists can also be stored using binary search trees. In a binary search tree, where the members are stored in a particular order depending on their values, we can insert a set of ordered elements. This makes it possible for us to quickly search for items in the list and carry out other tasks like determining the minimum or maximum element or the kth smallest item in the set (as described above). Binary search trees can also be used to hold dynamic sets, which allow for the addition and deletion of elements. We might need to do rotations and rebalancing operations to keep the tree height balanced in order to maintain the binary search tree's attributes. Conclusion To sum up, binary searches are essential to many algorithms and data structures, such as those that compute order statistics and store assorted lists. We can create more effective algorithms and data structures that can handle challenging computer science problems by comprehending the uses of binary searches.