1.10. Binary Search Tree Properties

A binary tree is just a tree with a root in which every node has either 1, 2, or 3 children, the heap poses a very special kind of binary tree, which we filled up top to bottom left to right, but in general a binary tree has no constraint.

A binary tree is a type of data structure that consists of nodes. Each node can have up to three children. While there are specific types of binary trees, such as heaps where nodes are filled in a particular order (top to bottom, left to right), a general binary tree does not have such constraints. This means that a binary tree can have a variety of structures.

Think of a binary tree like a family tree. In a family tree, each person (node) can have multiple children (other nodes). There isn’t a strict rule for how these children need to be arranged, similar to how in a binary tree, nodes can be placed in different structures.

We will assume that not only do we have a way to go from the parent to the child, but we also have a way to go from the child to the parent. So, every node we are going to assume will have a link to its parent, left child, and right child.

In a binary search tree, every node has connections to its parent as well as its children. Specifically, each node can point to its left child and right child. If a child does not have a child, that link is simply missing. This linking allows for easy navigation up and down the tree.

Imagine a tree in your backyard. Each branch (node) can connect to smaller twigs (children). If you want to go back up to the trunk (parent), you can easily see where each twig connects to the main branch.

In a binary search tree, the binary tree could have an arbitrary structure but the values are arranged in a specific way. For each node with a value v, all the nodes below v that are smaller than v are in the left subtree and all the values bigger than v are in the right subtree. And we typically will assume that there are no duplicate values.

The most important property of a binary search tree is how it organizes its values. For any given node containing a value 'v', all values in the left subtree are less than 'v' and all values in the right subtree are greater than 'v'. This strict ordering allows for efficient searching and insertion of values, as you can quickly disregard half the tree during lookups.

Think of a library that organizes books alphabetically. Each bookshelf holds books with titles starting with certain letters. If you are looking for a book with a title starting with 'H', you can skip all the shelves that contain books titled with 'A' through 'G', making it easier and faster to find the book.

So, 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 recursive traversal where we first walk through the left subtree, then the current node, and then the right subtree.

In-order traversal is a way of navigating through a binary search tree to retrieve values in sorted order. You start from the root of the tree, move to the leftmost node, print that value, then move up to the parent, print that value, and move to the right subtree. By following this pattern throughout the entire tree, you will gather all the values in ascending order.

Imagine sorting cards by their ranks. You go through the cards, first taking all cards lower than the one you're currently examining (left subtree), then you take that card and finally examine cards higher than it (right subtree). This ensures that when you're done, all cards are sorted.

Searching for a tree value in a binary search tree is very much like binary search. If the tree is empty, we say that it is not there, so we return false. If the current node has the value, then we have found it and return true. If we haven’t found it, then we look to see whether the value is smaller than the current value.

Binary search tree searches leverage its unique structure to find values efficiently. If the value you're searching for is less than the current node's value, you go left; if it's greater, you go right. This process continues until you find the value or reach a leaf node.

Think of playing a guessing game where you try to guess a number between 1 and 100. Each time you guess, the game tells you whether your guess is too high or too low, guiding you to narrow down your possibilities until you find the right number, just like traversing a binary search tree.

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