Required To Augment The Author's Binary Search Tree (BST) Co

Required To Augment The Authors Binary Search Tree Bst Code To Supp

Augment the Binary Search Tree (BST) class to support operations: nthElement(int n), rank(AnyType x), median(), isPerfect(), isComplete(), and toString(int nrLevels). These enhancements require adding a tree size attribute to each node for efficiency, and implementing recursive algorithms guided by this attribute.

Ensure the BST remains generic, prevents duplicates, and ignores removal of non-existent elements without unnecessary searches. The level-order print should output detailed node information with proper formatting and line breaks, reflecting the tree structure efficiently.

Paper For Above instruction

The task of augmenting a Binary Search Tree to provide advanced queries such as retrieving the n-th smallest element, calculating the rank of an element, and determining the median, as well as checking structural properties such as perfection and completeness, revolves around enhancing each node with additional data. This data—namely, the subtree size—serves as an essential guide for implementing these operations efficiently, avoiding costly traversals that could detract from performance.

Introduction

The Binary Search Tree (BST) is a fundamental data structure extensively utilized for efficient data storage and retrieval. However, basic BST operations such as insert, delete, and search do not directly support complex queries like rank retrieval or median identification. To address these limitations, augmentation techniques are employed, chiefly involving the storage of subtree sizes at each node. This enhancement enables algorithms to perform advanced operations in logarithmic time, particularly when the tree is balanced, or close to it. This paper discusses extending an existing generic BST class with additional methods, focusing on the implementation of the node augmentation and associated algorithms for the given operations.

Augmentation Strategy

The key to efficient advanced operations lies in the augmentation of the BST nodes with a treeSize attribute, representing the number of nodes in the subtree rooted at the current node, including itself. During insertions, deletions, or restructuring, this value must be updated appropriately, maintaining consistency across the tree. The recursive nature of these updates ensures minimal overhead while keeping the structure viable for complex queries.

Implementation of Methods

The core operations to be implemented are as follows:

1. nthElement(int n)

This method retrieves the n-th smallest element in the BST. Using the subtree sizes stored at each node, the algorithm compares n to the size of the left subtree, deciding whether to traverse left, right, or return the current node. If the left subtree size is L, then:

  • If n = L + 1, the current node is the n-th element.
  • If n ≤ L, recurse into the left subtree.
  • If n > L + 1, recurse into the right subtree with n- (L + 1).

2. rank(AnyType x)

This method finds the position of element x in an in-order traversal (starting from 1). Without calling find(), the algorithm navigates the tree, accumulating sizes of left subtrees and considering whether x is less than, greater, or equal to the current node's element. When x is found, the rank is computed as the size of the left subtree plus one plus the accumulated count.

3. median()

This method identifies the middle element based on the total size of the tree. If the total size is odd, the median is the ((size + 1)/2)-th element. If even, the median is the smaller of the two middle elements, i.e., the (size/2)-th element. Using nthElement() simplifies this implementation.

4. isPerfect()

A perfect binary tree is both full (all internal nodes have two children) and complete (all levels are fully filled). To determine this, the algorithm compares the total height of the tree with its size. Specifically, the height of a perfect binary tree is log2(size+1) - 1. Recursively, the check involves verifying that all leaf nodes are at the same depth, and all internal nodes have two children.

5. isComplete()

This property checks whether all levels, except possibly the last, are fully filled, and all nodes are as far left as possible. An algorithm for this is a level-order traversal that confirms whether any null child occurs before all nodes at that level are leaf nodes.

6. toString(int nrLevels)

This method produces a level-order string, printing nodes with their value, subtree size, and parent value. It uses a queue for level-order traversal, grouping output lines with no more than six nodes per line, and inserting blank lines between levels.

Efficient Algorithm Design

All these methods leverage subtree sizes to guide recursive or iterative traversal, ensuring minimal redundant searches. This approach employs the properties of BSTs and balanced traversal strategies; for instance, calculating kth elements or ranks proceeds in O(log n) time in balanced trees, with the size data enabling direct navigation.

Handling Duplicates and Non-Existent Elements

To maintain efficiency, duplicate insertions are ignored, and attempts to remove non-existent elements are also ignored, using the subtree sizes to streamline the process without preliminary searches. Recursive insert and remove methods are designed to correctly update all nodes’ subtree sizes during structural modifications.

Sample Implementation Snippet

In the insert method, after inserting a node, the subtree size is updated as:

 t.treeSize = 1 + size(t.left) + size(t.right);

Similarly, in removal and restructuring, subtree sizes are recalculated upon return from recursive calls. The size helper function returns 0 if the node is null.

Conclusion

Augmenting BST nodes with subtree sizes significantly enhances the data structure, enabling complex queries with high efficiency. Proper recursive algorithms, combined with accurate maintenance of the size attribute during insertions and deletions, are crucial for performance. These enhancements expand the utility of the BST in applications requiring ranked, median, and structural property queries with minimal computational overhead.

References

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