16.2.3. Deleting a Node with Two Children

So, how do we delete a node? We are given a value v, and we must find v in the tree to delete the node containing it.

To delete a node in a binary search tree, you first need to find the node that contains the value you want to delete by traversing the tree. This involves checking whether to move left or right based on the value at the current node until you locate the node with the value to delete.

Imagine you're searching for a book in a library. You start at the information desk (the root of the tree) and decide which aisle (left or right) to go down based on the book's title. You continue checking books (nodes) until you find the one you need (the node with the value).

The simplest case to handle is when the node is a leaf node. For instance, if you want to delete 65, we search for 65 and find it as a leaf node. We can just remove this node from the tree, and it remains a valid search tree.

When a node is a leaf (it has no children), deletion is straightforward. Simply remove the node from its parent by setting the appropriate pointer (left or right) to null. This ensures that the structure of the tree remains valid.

Think of a leaf node like a book that is placed on the end of a shelf. If you decide to remove the book, you simply take it away, and the shelf (the tree) remains intact—there are no gaps or missing sections.

Sometimes, a deleted node might have only one child. For instance, if we delete 74, which has a right child, we can promote this child by connecting the parent directly to the right child.

If the node to be deleted has only one child, you delete the node and link its parent directly to the child. This effectively removes the node from the tree without disrupting its structure since the child takes its place.

Imagine you have a plant that has grown too large for its pot. If you need to remove it from the pot (the node), instead of just throwing it away, you can directly connect the plant’s roots (the child) to the soil outside the pot (the parent's new connection).

What if the node you want to delete has two children? In this case, we cannot simply remove it, as it has connections we must maintain to uphold the binary search tree properties.

When faced with a node that has two children, the node must be replaced with either its predecessor (the maximum value in the left subtree) or its successor (the minimum value in the right subtree) to maintain the binary search tree structure after deletion.

Think of a two-child node like a manager of a department (the node) who has two assistants (the children). If the manager leaves (is deleted), you need to promote one of the assistants to take over the role while ensuring the department continues to function effectively.

To delete the node with two children, we first compute the predecessor node, which will replace the current node's value, and then delete the predecessor.

The predecessor is identified by the largest value in the left subtree of the node to be deleted. Once this value is found, you replace the value of the original node with that of the predecessor and then call the delete operation again to remove the predecessor node.

Consider that after promoting an assistant to manager, you need to fill the assistant's position. You might look for someone else in the team who is best suited for the job, promoting them while allowing the previous assistant to leave (be deleted) without leaving gaps in responsibilities.

After replacing the value with the predecessor's value, you delete the predecessor, which simplifies the case as it will either be a leaf or have one child.

Once the predecessor's value replaces the original node's value, the tree structure remains valid because the order of nodes is preserved. The predecessor being a leaf or having a single child allows for a straightforward deletion of that node.

If a team member leaves (the predecessor), the new manager can simply take over their responsibilities without disrupting the team's workflow, ensuring everything continues running smoothly.

All these operations tend to work down a single path. Therefore, the worst-case time complexity for each operation is proportional to the height of the current tree.

The efficiency of searching for a node to delete directly relates to the tree's height. In a balanced tree, operations can be performed in logarithmic time, making them efficient even as the number of nodes increases.

Think of searching through a large library. If the books are organized neatly by genre, it’s quicker to find what you're looking for than if they’re scattered all over the place. Similarly, a balanced tree allows for faster search times, just like organized books.

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