How do you insert into a binary search tree?

To insert into a binary search tree, you compare the new node with the root and place it in the appropriate subtree.

In a binary search tree (BST), each node has a unique key, and for each node, all keys in its left subtree are smaller, and all keys in its right subtree are larger. When inserting a new node, you start from the root and compare the new node's key with the root's key. If the new key is smaller, you move to the left child, and if it's larger, you move to the right child. You continue this process until you find an empty spot where the new node should be inserted.

Let's consider an example. Suppose you have a BST with root key 8, and you want to insert a new node with key 3. You start from the root. Since 3 is less than 8, you move to the left child. If the left child is null, you insert the new node there. If not, you compare the new key with the left child's key and decide whether to move to its left or right child. You repeat this process until you find a null spot to insert the new node.

The time complexity of the insertion operation in a BST is O(h), where h is the height of the tree. In the worst case, when the tree is skewed, the height of the tree is equal to the number of nodes, and the time complexity becomes O(n). However, in the average case, where the tree is balanced, the height is log(n), and the time complexity is O(log n).

In conclusion, inserting into a binary search tree involves a series of comparisons starting from the root. The new node is placed in the correct position to maintain the BST property. This operation is efficient, with a time complexity of O(log n) in the average case.