How to determine the time complexity of inserting a node into a binary tree?

The time complexity of inserting a node into a binary tree depends on the type of tree and its balance.

• In the worst case (for example, in an unbalanced binary search tree where the tree degenerates into a list), the complexity will be O(n), where n is the number of nodes.
• In the average and best case for a balanced binary tree (such as AVL or red-black tree), the insertion complexity is O(log n), since the height of the tree is proportional to the logarithm of the number of nodes.

Therefore, to determine the time complexity, one must consider the structure of the tree and its balancing.