Question

Insert 28,25,26,42,47,30,45,29,5 into an initially empty binary search tree. Draw the final binary search tree.

Short answer

The final binary search tree is created by placing each subsequent number based on binary search rules.

Step-by-step solution

Insert the First Element

Begin with an empty binary search tree (BST). Insert the first number, 28. As the tree is empty, 28 becomes the root node.

Insert 25

Compare 25 with the root node 28. Since 25 is less than 28, insert 25 as the left child of 28.

Insert 26

Compare 26 with 28 and 25. Since 26 is less than 28 but greater than 25, insert 26 as the right child of 25.

Insert 42

Compare 42 with the root node 28. Since 42 is greater than 28, insert 42 as the right child of 28.

Insert 47

Compare 47 with 28 and 42. Since 47 is greater than both, insert 47 as the right child of 42.

Insert 30

Compare 30 with 28 and 42. Since 30 is greater than 28 but less than 42, insert 30 as the left child of 42.

Insert 45

Compare 45 with 28, 42, and 47. Since 45 is greater than 28 and 42 but less than 47, insert 45 as the left child of 47.

Insert 29

Compare 29 with 28, 42, and 30 in that order. Since 29 is greater than 28 but less than both 42 and 30, insert 29 as the left child of 30.

Insert 5

Compare 5 with 28 and 25. Since 5 is less than both, insert 5 as the left child of 25.

Key concepts

Tree Traversal

Tree traversal in binary search trees (BSTs) involves visiting nodes in a specific order to retrieve or manipulate data efficiently. There are several types of traversal techniques:

• Inorder Traversal: This method visits nodes in ascending order. It first visits the left subtree, then the root, and finally the right subtree. This is particularly useful for BSTs as it returns nodes in a sorted sequence.
• Preorder Traversal: Here, the traversal visits the root node first, then recursively visits the left subtree followed by the right subtree. This is useful for creating a copy of the tree.
• Postorder Traversal: This approach visits the left subtree first, then the right subtree, and the root last. This is often used for deleting a tree.
• Level Order Traversal: Also known as breadth-first traversal, this technique visits nodes level by level from top to bottom, usually using a queue data structure.

Each traversal method has its own unique application and choosing the right one can optimize operations performed on the BST.

Node Comparison

Node comparison is a fundamental operation when inserting elements into a BST. It involves determining the position of a new node by comparing it with existing nodes, starting from the root.
When inserting, we compare:

• If the value is less: Move to the left child. In our example, inserting 25 meant comparing with 28 and moving left because 25 is less.
• If the value is greater: Move to the right child. For instance, inserting 42 required moving right from 28 because 42 is greater.

By repeatedly comparing nodes, we maintain the BST structure that allows for efficient data retrieval. Correct comparisons ensure that each node's left child has lesser value, and the right child has a greater value than the node itself.

Data Structures

Binary Search Trees are a specialized type of data structure built from simple data structures like nodes and pointers.
Each node in a BST contains:

• A data element (e.g., a number like 28 or 42).
• Pointers or references to child nodes (left and right).

BSTs efficiently support operations such as insertion, deletion, and search due to their structured layout, which reduces the need for redundant searches.
They are part of a broader category called hierarchical data structures, which also include structures like heaps and binary trees, each having unique properties and use-cases that fit different problem-solving scenarios.

BST Properties

Binary Search Trees hold several properties that make them valuable in computing:

• Ordered Structure: Nodes are arranged such that for any given node, its left child will always have a smaller value, and the right child will have a greater value. This makes searching, adding, or removing elements highly efficient.
• Dynamic Data Type: Unlike arrays, BSTs can grow and shrink dynamically. This flexibility is useful for applications where the number of elements is unpredictable.
• Efficient CRUD Operations: With a well-balanced tree, CRUD (Create, Read, Update, Delete) operations generally take \( ext{O}(\log n)\) time, making them faster in comparison to linear data structures.

Understanding these properties helps in leveraging BSTs optimally depending on your specific use case and can also inform decisions on whether a BST is appropriate compared to other data structures.