Question

Give an algorithm for a function int largest (TreeNode *tree) that takes a pointer to a root of a binary search tree as parameter and returns the largest value stored in the tree.

Short answer

**Question**: Create an algorithm for a function that takes a binary search tree as input and returns the largest value stored in the tree. **Solution**: We can implement the solution either iteratively or recursively by following the right child pointers from the root node until we reach a node with no right child, which will contain the largest value. Iterative Solution: ```cpp int largest (TreeNode *tree) { TreeNode *current_node = tree; while (current_node->right != nullptr) { current_node = current_node->right; } return current_node->value; } ``` Recursive Solution: ```cpp int largest (TreeNode *tree) { if (tree->right == nullptr) { return tree->value; } return largest(tree->right); } ```

Step-by-step solution

Function signature

We start by writing the function signature as given in the problem statement: ```cpp int largest (TreeNode *tree); ``` The function `largest` takes a pointer to the root node of a binary search tree as an argument and returns an integer which is the largest value in the tree.

Iterative solution

In this step, we will implement the algorithm iteratively. We start from the root node and follow the right subtree until we find a node with no right child: ```cpp int largest (TreeNode *tree) { TreeNode *current_node = tree; while (current_node->right != nullptr) { current_node = current_node->right; } return current_node->value; } ``` In this implementation, we initialize a pointer `current_node` that initially points to the root of the tree. Then, we use a while loop to traverse the tree from the root to the node with the largest value. Inside the loop, we check if the right child of the current node is not `nullptr`. If so, we update the `current_node` pointer to point to its right child. We continue doing this until we reach a node with no right child. Finally, we return the value of the node containing the largest value.

Recursive solution

Alternatively, we can solve this problem using a recursive approach. We will call the function recursively if the right child of the current node exists, otherwise, we will return the value of the current node: ```cpp int largest (TreeNode *tree) { if (tree->right == nullptr) { return tree->value; } return largest(tree->right); } ``` In this implementation, we check if the right child of the current node is `nullptr`. If so, we return the value of the current node since it contains the largest value in the tree. Otherwise, we call the `largest` function recursively with the right child of the current node as its argument.

Key concepts

Binary Search Tree

A Binary Search Tree (BST) is a type of data structure that efficiently organizes elements for quick search, insertion, and deletion. Each node in a BST has:

• A value.
• A pointer to a left child that contains values less than the node's value.
• A pointer to a right child that contains values greater than the node's value.

The key characteristic of a BST is that it maintains this ordered structure, which facilitates efficient operations like searching for elements. Suppose you have a BST and you want to locate the largest value. You can achieve this by traversing to the rightmost node, as larger values are stored to the right. This is the core concept we explore when finding the largest value in a BST.

Algorithm Design

When designing an algorithm, the goal is to create a step-by-step procedure that efficiently solves a problem. Good algorithm design considers both the clarity of the solution and its efficiency. Let's focus on finding the largest value in a binary search tree.
The algorithm needs to systematically navigate the BST, ensuring every step aligns with the desired outcome. Two common approaches are iterative and recursive methods, each with their benefits.
Iterative solutions tend to be more memory efficient while avoiding the potential pitfalls of excessive function call stacks that occur with recursion. For this problem, it involves following the right child pointers until a leaf node is reached, which holds the largest value.

Recursion

Recursion is a fundamental programming technique where a function calls itself to solve smaller instances of the problem. In the context of our problem, recursion can identify the largest value in a binary search tree by repeatedly calling the same logic on the tree's right child.
The recursive solution provided here checks if a node's right child exists. If it does, the function calls itself with this right child, continuing until a node with no right child is found. This node holds the largest value, which is then returned as the recursion unwinds.

• Recursion often simplifies code and can make it more readable, but it may introduce additional overhead due to repeated function calls.
• Stack overflow is a risk for very deep recursive calls, although this is rare in balanced trees.

Recursion showcases beautifully how elegant solutions can be derived by breaking down a problem into simpler, repetitive tasks, mirroring the natural divide-and-conquer approach.

Iterative vs Recursive

When implementing solutions, developers often face the decision of choosing between iterative and recursive approaches.
In our specific task of finding the largest node in a binary search tree, both methods were presented. Each offers distinct advantages and challenges.

• Iterative Approach: This method employs loops to navigate the tree. It is memory-efficient and removes the risk of stack overflow from excessive recursive calls. In our case, a simple while loop traverses to the rightmost node effectively.
• Recursive Approach: More readable due to its simple, logical flow, recursively calling the function on the node's right child until the largest child is found. However, it can be less performant with each call consuming stack memory.

Choosing between these relies on factors such as memory constraints, performance needs, and readability preferences. While recursive solutions are often more intuitive, the iterative alternative might be chosen for its efficiency and stability, especially in environments where recursion depth could grow too large.