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Chapter 8: Problem 12

Write an algorithm that deletes a node from a binary search tree considering all possible cases. Analyze your algorithm, and show the results using order notation.

Short Answer

Expert verified
The algorithm to delete a node from a binary search tree takes into account three cases: when the node is a leaf, when it has only one child, and when it has two children. The time complexity, in the worst-case scenario, is O(n) (skewed tree), whereas for a balanced BST, the time complexity is O(h), h being the height of the tree.

Step by step solution

01

Define the Problem in Detail

Firstly, you must fully understand the properties of a binary search tree - for any given node, every value of the left child node and its descendants are less than the value of the given node and every value of the right child node and its descendants are higher. The problem here is to implement an algorithm to delete a node considering all possibilities, including when it is a leaf node, a node with one child, or a node with two children.
02

Deleting a Leaf Node

If the node to delete is a leaf (node with no children), then it is the most straightforward case. Simply remove the node from the tree.
03

Deleting a Node with One Child

If the node to delete has only one child, replace the node by its child and remove the node.
04

Deleting a Node with Two Children

The process to delete a node with two children is more complicated. The trick here is to find a replacement for the node from either its in-order predecessor (rightmost node of left subtree) or its in-order successor (leftmost node of right subtree) since they respectively are the nearest lesser and greater values. Replace the node with the in-order predecessor (or successor) and delete that predecessor (or successor). If this in-order predecessor (or successor) has a left child then this child becomes the right child of the predecessor's (or successor's) parent.
05

Time Complexity Analysis

In the worst case scenario (tree being a skewed tree), the time complexity could be O(n) where n is the number of nodes. But for a balanced binary search tree, the time complexity would be O(h) where h is the height of the tree, as we traverse from root to leaf node. The deletion itself takes O(1) time.

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Most popular questions from this chapter

Show that the average-case time complexity of Interpolation Search is in \(\Theta(\lg (\lg n)),\) assuming the keys are uniformly distributed and that search key \(x\) is equally probable to be in cach of the array slots.

Write a probabilistic algorithm that determines whether an array of \(n\) elements has a majority element (the element that appears the most). Analyze your algorithm, and show the results using order notation.

Discuss the advantages of using each of the following instead of the other: (a) A binary search tree with a balancing mechanism (b) \(A 3-2\) tree

Let \(S\) and \(T\) be two arrays of \(n\) numbers that are already in nondecreasing order. Write an algorithm that finds the median of all \(2 n\) numbers whose time complexity is in \(\Theta(\lg n)\)

Write an algorithm that lists all the keys in a \(3-2\) tree in their natural order. Analyze your algorithm, and show the results using order notation.
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