Question

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

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

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.

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.

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.

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.

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.

Key concepts

Algorithm Design

Designing an algorithm for a binary search tree (BST) involves understanding its structure. A BST is organized such that each node has a value, and all values in the left subtree are less than the node's value, while all values in the right subtree are greater. This property allows for efficient searching, inserting, and deleting operations.

When deleting a node in a BST, several scenarios need consideration. You might delete a leaf node, which has no children, a node with one child, or a node with two children. Each scenario demands a different approach:

• **Leaf Node**: Directly remove it.
• **One Child**: Replace the node with its child.
• **Two Children**: Find either the in-order predecessor or successor, replace it, and adjust the tree structure.

This structured approach ensures that the tree maintains its properties after deletion.

Node Deletion

Node deletion in a binary search tree must handle three possible cases to maintain the BST properties. These are detailed as follows:

**Deleting a Leaf Node**: This is the simplest case. Since the node has no children, removing it doesn't affect the rest of the tree. Simply unlink the node from its parent.

**Deleting a Node with One Child**: If the node to be deleted has either a left or right child, replace the node with its child. This involves changing the parent of the node to point to the child directly.

**Deleting a Node with Two Children**: This is more complex. First, locate the node's in-order successor (the smallest node in the right subtree) or in-order predecessor (the largest in the left subtree). Replace the node's value with this successor or predecessor, and then delete this node, which is now a simpler case since it will have at most one child.

Time Complexity Analysis

Analyzing the time complexity of BST operations requires understanding how work is distributed in the tree. The time to perform any operation in a BST is generally proportional to the tree's height.

**Balanced Trees**: If the tree is balanced, the height is approximately \log(n)\, where \(n\) is the number of nodes. Thus, deletion operations have time complexity \(O(\log n)\).

**Skewed Trees**: In a worst-case scenario, where the tree is skewed (like a linked list), the height becomes \(n\), and the complexity for deletion rises to \(O(n)\).

Understanding this analysis helps in recognizing the efficiency of operations under different tree structures. Keeping the tree balanced ensures optimal performance.

Order Notation

Order notation is a mathematical notation to describe the limiting behavior of a function when its parameters tend towards a particular value or infinity. It's essential in analyzing algorithm efficiency.

For a binary search tree node deletion algorithm, the time complexity, expressed using big O notation, gives an upper bound on the time requirement as the input size grows.

• **\(O(\log n)\)**: Represents operations in a balanced tree where the height is proportional to \(\log n\).
• **\(O(n)\)**: Represents operations in a skewed tree, where the tree's height is \(n\).

Order notation provides a high-level view of an algorithm's performance, enabling comparisons and optimizations. Understanding it allows developers to assess how changes to input size will affect time requirements.