1.11. In-order Traversal

The first thing that we can do with a binary search tree is to list out its values in sorted order, this is called an in-order traversal.

In-order traversal is a method of visiting each node in a binary search tree (BST). It allows us to retrieve the values of the tree in sorted order. The method follows a specific approach: for each node, we first visit the left subtree, then process the current node, and finally, we visit the right subtree.

Imagine you are organizing a bookshelf. You start by looking on the leftmost shelf for books to process first (left subtree), then you check the shelf you are currently on (current node), and finally, you look to the rightmost shelf (right subtree). By doing this, you ensure that all books get organized in a neat, alphabetical order.

So, in order traversal is recursive traversal it is a way of walking through the tree. So, that we first walk through the left sub tree, then the current node and then the right sub tree.

In an in-order traversal, we leverage recursion to accomplish the task. For each node in the tree, we recursively perform the traversal on its left child, which means reaching the end of the leftmost path of the tree before handling the value of that node. Once we’ve processed the left child completely, we ‘take a moment’ to handle the current node, and finally, we proceed to the right child. This process ensures that we access all the nodes in a sorted manner.

Think of a tree as a family reunion where everyone is supposed to introduce themselves. You first make sure to greet all the relatives on your left side (left subtree), then you acknowledge the person at the middle (current node), and finally you move to the right side of the gathering (right subtree). This way, you’ve given everyone a chance to introduce themselves while maintaining order.

For example, if you perform an in-order traversal here, we start at the root and then we have to first walk on the left subtree, so we walk to the left child.

When executing an in-order traversal, we begin at the root of the binary search tree. Our first action is to explore the left subtree. If there is a node without a left child, we output the value of that node, and then backtrack to print the value of the parent node and move rightwards. This continues until all nodes have been visited and their values printed in sorted order.

Imagine you are searching through a library's collection by starting from the leftmost end and moving to the right. As you find each book (node), if it's the last on the left, you read it (print it out), and then you check the next book on the shelf (the parent). If that book has a right neighbor, you go there next. This way, you always move through the collection sequentially.

So, it is easy to see that because of this property, we know that everything to the left is smaller than that value and everything to the right is bigger than the value.

The nature of binary search trees ensures that in an in-order traversal, we attain a sorted output. Since all nodes to the left of any given node are less than that node and all nodes to the right are greater, following the in-order traversal will yield a sequence of values starting from the smallest to the largest.

Consider the process of sorting a stack of cards. Each time you pick a card from the left side of the stack (smaller values) and set them down in order, you're effectively sorting them as you proceed. Every time you pick a card that belongs on the right, you automatically ensure it's larger than those you've already placed down, achieving a sorted sequence.

So, now searching for a value in a binary search tree is very much like binary search.

When searching for a value in a binary search tree, the process mirrors binary search. We start at the root and compare the target value with the current node's value. If it matches, we have found our value. If the target is smaller, we focus our search on the left subtree; if larger, we move to the right subtree. This efficient approach is a direct result of the properties of BST and allows for quick lookups.

Think of searching for a book in a library catalog where you know books are arranged by their titles. You narrow down your search: if your desired title comes before the current title you're comparing with, you check the left section (left subtree). If it comes after, you check the right section (right subtree). This systematic approach helps you find the book quickly without having to browse through every single title.

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