Binary Tree Operations

Binary trees store data in a hierarchical structure. Unlike binary search trees (BSTs), they do not follow any ordering rules, so we cannot directly determine where to go when searching or inserting.

Because of this, most operations rely on traversing the tree.

In this lesson, we study three fundamental operations:

• Searching

• Insertion

• Deletion

Searching in a binary tree

Searching in a binary tree means checking whether a given value exists in the tree. As there is no ordering property, we cannot skip parts of the tree. In the worst case, we may need to visit every node.

A common approach is to use depth-first search (DFS). We start at the root, check its value, and if it does not match, we recursively search the left subtree and then the right subtree.

How this algorithm works

1. Start at the root node.

2. If the current node is None, return False because the value is not found.

3. Check if the current node’s value matches the target:

   1. If yes, return True

4. If not, recursively search the left subtree.

5. If the value is not found in the left subtree, search the right subtree.

6. Return True if the value is found in either subtree, otherwise return False.

Python implementation

Below is the Python implementation of search in a binary tree.

Time complexity: $O(n)$ ...