Question

Are short-answer questions. Draw the binary search tree whose elements are inserted in the following order: \(\begin{array}{ccccccccc}50 & 72 & 96 & 107 & 26 & 12 & 11 & 9 & 2 \\ 10 & 25 & 51 & 16 & 17 & 95 & & & \end{array}\)

Short answer

The binary search tree is rooted at 50, with 26 left, 12 left of 26, 11 left of 12, 9 left of 11, 2 left of 9, 10 right of 9, 25 right of 12, 16 left of 25, 17 right of 16, 72 right of 50, 51 left of 72, 96 right of 72, 95 left of 96, and 107 right of 96.

Step-by-step solution

Insert the root

Start by inserting the first element of the sequence, 50, as the root of the binary search tree (BST).

Insert 72 - Greater than root

Compare the next element, 72, with the root. Since 72 is greater than 50, insert 72 as the right child of 50.

Insert 96 - Greater than 72

Compare 96 with the root, then with 72. As 96 is greater than both, place it as the right child of 72.

Insert 107 - Greater than 96

Compare 107 with 50, 72, and finally 96. Since it's greater than all of them, place 107 as the right child of 96.

Insert 26 - Smaller than root

Compare 26 with the root. Since 26 is less than 50, insert it as the left child of 50.

Insert 12 - Smaller than 26

Compare 12 with 50 and then with 26. Since 12 is less than both, place 12 as the left child of 26.

Insert 11 - Smaller than 12

Compare 11 with 50, 26, and 12. Since it's smaller than all, insert 11 as the left child of 12.

Insert 9 - Smaller than 11

Compare 9 with each node on the path from 50 to 11. Since 9 is smaller than all, insert it as the left child of 11.

Insert 2 - Smaller than 9

Compare 2 through the path from 50 to 9. Since 2 is smaller, insert it as the left child of 9.

Insert 10 - Between 9 and 11

Insert 10 by tracing the pathway. It fits between 9 and 11, so place 10 as the right child of 9.

Insert 25 - Between 12 and 26

Through comparisons, place 25 between 12 and 26, making it the right child of 12.

Insert 51 - Between 50 and 72

Locate 51 between 50 and 72, inserting it as the left child of 72.

Insert 16 - Between 12 and 25

Find the appropriate position for 16 as the left child of 25, using the comparisons route from the root.

Insert 17 - Greater than 16

Place 17 as the right child of 16, being greater than 16 but less than 25.

Insert 95 - Between 72 and 96

Compare 95 along the path and determine it fits as the left child of 96, completing the tree.

Key concepts

Tree Insertion

Inserting elements into a Binary Search Tree (BST) involves placing each element from the sequence into its correct position. It is based on comparisons with current nodes. Here's how it works:

• Start with the Root: The first element, 50, becomes the root. This is the starting point of any BST.
• Compare and Insert: Each subsequent element is compared with the current nodes to find its rightful place.
  • If an element is less than the current node, move to the left child.
  • If greater, move to the right child.
• Placement: Once a correct spot is found (where a child is empty), the element gets inserted.

By following these steps, each element is systematically inserted, maintaining the BST properties, where all left children are less, and right children are greater than the current node. The process repeats until all elements are placed.

Binary Tree Construction

Constructing a binary tree might seem simple at first, but it requires following specific rules to ensure that the binary search tree remains valid. Let's break this down:

• Starting with the Root Node: The binary tree construction begins by inserting the very first element as the root node, setting the foundation for the rest of the tree.
• Building the Branches: As each new element is added, it becomes a new branch based on comparisons with existing elements in the tree.
  • Elements less than a node are consistently placed on the left.
  • Elements greater than a node are consistently placed on the right.
• Balancing Act: While the binary tree structure may become unbalanced depending on the order of insertion, the rules ensure that similar elements are organized in a natural hierarchical order.

This process continues until all elements in the sequence have been appropriately added, forming a structured and navigable binary tree.

Data Structures

A Binary Search Tree (BST) is a vital example of a dynamic data structure, which is adaptable for varied computational needs. Here’s why BSTs are essential in data structures:

• Hierarchical Organization: BSTs store data in a hierarchical manner, allowing quick search, addition, and deletion operations.
• Dynamic Flexibility: Unlike static arrays, BSTs grow and shrink dynamically as elements are inserted or removed.
• Ordered Structure: The maintenance of order through the left-child, right-child rule enables efficient searching.
  • By design, in-order traversal of a BST yields sorted data.
• Key Operations: BSTs support key operations such as searching, insertion, and deletion efficiently, typically in logarithmic time in balanced trees.

Thus, a BST provides a structured way to store and manage data efficiently, making them an invaluable tool in many algorithms and systems.

Algorithms

The algorithm behind constructing a Binary Search Tree is systematic and relies heavily on comparison operations. Here’s a simple breakdown:

• Step-by-Step Comparison: Each element is iteratively compared starting from the root down the tree.
  • The decision to move left or right is based on whether the new element is less or greater than the current node.
• Element Placement: Once a spot is found where the comparator node’s child is null, the element is inserted as that child.
• Efficiency and Complexity: For a perfectly balanced BST, the complexity of search, insert, and delete operations is logarithmic, (\(O(\log n)\)). However, the structure can degrade into a linked list in skewed cases, resulting in linear complexity, (\(O(n)\)).

The algorithm is both simple and powerful, making it suitable for use in various applications such as databases, computation engines, and memory management systems.