Question

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

Short answer

Answer: The 'smallest' function is used to find and return the smallest value stored in a binary search tree. It does this by traversing through the left subtree of the tree, since the left subtree contains values less than the root, until it reaches a leaf node.

Step-by-step solution

Function Declaration and Base Case

Declare the function 'int smallest(TreeNode *tree)' and check for the base case. If the tree is null, there is no smallest value, so return INT_MAX (the maximum value an int can hold). ```cpp #include int smallest(TreeNode *tree) { if (tree == nullptr) { return INT_MAX; } ```

Traverse Left Subtree

Traverse the left subtree by calling the 'smallest' function recursively, until reaching a leaf node. ```cpp int left_smallest = smallest(tree->left); ```

Compare Node Values

Compare the current node's value with the smallest value found in the left subtree. Return the smaller value. ```cpp return min(tree->value, left_smallest); } ``` Now that the algorithm is complete, here's the final code: ```cpp #include int smallest(TreeNode *tree) { if (tree == nullptr) { return INT_MAX; } int left_smallest = smallest(tree->left); return min(tree->value, left_smallest); } ```

Key concepts

Binary Search Tree

A Binary Search Tree (BST) is a special type of binary tree. It organizes data in a way that allows for efficient searching, insertion, and deletion operations. In a BST, each node has a value, and:

• The left subtree of a node will only contain values less than that node's value.
• The right subtree of a node holds values greater than or equal to the node's value.
• Both the left and right subtrees are themselves binary search trees.

This structure makes searching through a BST an efficient process. In an average case scenario, operations like search, insertion, and deletion take logarithmic time, \(O(\log n)\). However, if the tree becomes unbalanced, the operations might degrade to linear time, \(O(n)\), like in a linked list.
Given that the smallest value in a BST would be located on the far-left side, the algorithm to find the smallest element takes advantage of this structure by recursively visiting left children until it reaches a null pointer. Understanding BSTs is crucial when we need to design algorithms for efficiently managing sorted data.

Tree Traversal

Tree traversal is a method to visit each node in a tree data structure, processing the data in some systematic way. There are several types of tree traversal methods, each serving different purposes.

• In-order Traversal: In a BST, in-order traversal visits nodes in the ascending order of their values. First, it visits the left child, then the node itself, and finally the right child.
• Pre-order Traversal: This involves visiting the node first, followed by the left subtree, then the right subtree. It is often used to create a copy of the tree.
• Post-order Traversal: Nodes are visited in this order: left subtree, right subtree, and the node itself. This is useful for deleting the tree, as it removes children before the parent.

In the context of the provided exercise, understanding in-order traversal is helpful because it relates to finding the smallest and largest elements in a BST. By only traversing the left subtree, as the task requires, the algorithm can efficiently find the smallest element without needing to look at other parts of the tree.

Recursion

Recursion is a powerful programming technique where a function calls itself to solve a smaller instance of the same problem. It can be visualized as a function breaking down a larger problem into subproblems until they reach a simple base case that can be solved directly.
In the context of the 'smallest' function, recursion allows the function to continuously call itself with the left child of the tree until a terminal node (a leaf or null pointer) is reached. This approach makes the algorithm concise and clean, as it automatically manages the iteration through the tree nodes:

• The base case here is when the function encounters a null pointer, indicating that it has reached beyond the leaf node, at which point it returns \({\text{INT\_MAX}}\).
• By using recursion, the code remains simple. There is no need to manually loop through the elements of the tree; the recursive calls handle this as they "unwind."

Recursion is ideal for problems like tree traversal where the nature of the problem naturally fits the recursive pattern.

Algorithm Design

Algorithm design involves the process of creating algorithms that not only solve the given problem but do so in an efficient and understandable manner. It is about organizing the process in such a way that the solution is both correct and performs well in terms of time and space.
When designing algorithms like the one in the exercise, several considerations must be taken into account:

• Correctness: Ensuring that the algorithm accurately identifies the smallest element in all cases, including edge cases where the tree might be empty or have only one node.
• Efficiency: The algorithm takes advantage of the BST structure, offering a \(O(h)\) complexity, where \(h\) is the height of the tree, by only processing once through the leftmost nodes.
• Simplicity: By leveraging recursion, the code remains straightforward and easier to understand, reducing the complexity of manually managing stack operations that would be needed in an iterative solution.

Algorithm design is crucial in computer science as it lies at the heart of developing software that can perform operations quickly and correctly, which is essential in all applications of data structures like binary search trees.