Question

Suppose the following values are inserted into a binary search tree, in the order given: \\[12,7,9,10,22,24,30,18,3,14,20\\] Draw a diagram of the resulting binary tree.

Short answer

Question: Draw the binary search tree resulting from inserting the given list of values: [12, 7, 9, 10, 22, 24, 30, 18, 3, 14, 20] Answer: The binary search tree after inserting all values will look like: ``` 12 / \ 7 22 / \ / \ 3 9 18 24 \ / \ 10 14 30 / 20 ```

Step-by-step solution

Initialize an empty binary search tree

An empty binary search tree is initialized before inserting any value.

Insert the first value (12)

As the binary search tree is empty, we insert the first value 12 as the root node.

Insert the next value (7)

Next, we compare the value 7 with the root node value 12. Since 7 is less than 12, it is inserted as the left child of root node 12.

Insert the next value (9)

Now we compare the value 9 with the root node value 12, as 9 is less than 12, we compare it with the left child node value 7. Since 9 is greater than 7, it is inserted as the right child of the node 7.

Insert the next value (10)

Similarly, we compare the value 10 with the first three nodes as follows: \\ - Compare 10 with root node (12). 10 is less than 12, so we proceed to the left child (7). \\ - Compare 10 with the left child node (7). 10 is greater than 7, so we proceed to the right child of 7 (9). \\ - Compare 10 with the right child node (9). 10 is greater than 9, so we insert 10 as the right child of the node 9. Continue inserting the remaining values using the same procedure.

Insert the rest of the values (22, 24, 30, 18, 3, 14, 20)

Follow the same method of comparing the values and inserting them to the appropriate position based on the binary search tree property. The resulting binary search tree will have the following structure: ``` 12 / \ 7 22 / \ / \ 3 9 18 24 \ / \ 10 14 30 / 20 ``` By following these steps, we have successfully drawn the diagram representing the binary search tree for the given list of values.

Key concepts

Tree Data Structure

At the heart of many complex and efficient data operations lies the tree data structure. A tree is an abstract model of a hierarchical structure that consists of nodes with a parent-child relationship. Trees are similar to a real-life tree, with branches spreading out from a single trunk; only in data structures, they are represented upside down, with the root at the top.

Each node in a tree contains data and may have a left and right child, representing two subtrees.
Trees are non-linear and can reflect real-world structures such as company organization charts or file systems. A special kind of tree, called a 'binary search tree' (BST), allows for incredibly efficient data storage, retrieval, and manipulation. In a BST, each node can have at most two children, and the left child must contain a value lesser than its parent, whereas the right child contains a greater value. This property enables operations such as search, insertion, and deletion to be performed with an average time complexity of O(log n) if the tree is balanced.

Binary Tree Insertion

The process of adding new nodes to a binary search tree is straightforward but must obey the BST properties. The steps to inserting a new node into a BST are logically determined by its value in order to maintain the ordering of the elements.

Here’s a simplified view of how it is generally done:

• Start at the root node.
• If the BST is empty, the new node becomes the root.
• If the new node's value is less than the current node's, move to the left child.
• If the new node's value is greater than the current node's, move to the right child.
• Repeat the process until the correct position is found.
• Insert the new node as a leaf in the found position.

By following these rules, the BST maintains its integrity, allowing for efficient searching for a value by systematically narrowing down the possible locations through comparison, significantly speeding up the process compared to linear data structures.

C++ Programming

C++ is a powerful, high-performance programming language widely used for system/software development and game programming. It's known for enabling direct manipulation of hardware resources and providing fine-grained control over memory management.

Implementing a binary search tree in C++ is a great exercise for understanding object-oriented programming and pointers. Here are the fundamental steps you would follow in implementing a binary search tree in C++:

• Define a 'Node' structure with data, a pointer to the left child, and a pointer to the right child.
• Create a 'BST' class, which will contain a Node pointer to the root and public methods for operations like 'insert', 'search', and 'delete'.
• Implement the 'insert' method using recursion or iteration based on the aforementioned binary tree insertion logic.
• Provide additional functionalities, such as traversal methods to navigate and display the tree structure.

In C++, the combination of classes, pointers, and recursion can elegantly handle the complexities of a binary search tree, making it an excellent topic for students to deepen their understanding of algorithms and data structures.