16.2. Deletion in a Search Tree

So, how do we delete a node, with delete says, but we given a value v which we find v in the tree, you must delete the node containing. So, the basic case that is very simple to handle is one the node is a leaf node, because then we can just delete it and then it just falls off the tree at the bottom. So, for instance if you want to delete 65, then we will search for 65 find it by the usual mechanism of following the left and right paths. And then since it is a leaf node we can just remove this node from the tree and nothing happens, it remains a valid search.

The deletion of a node in a search tree can vary based on the node's position and its children. The simplest case is when the node to be deleted is a leaf node (a node without children). When you want to delete a leaf node, such as node 65, you simply search for it in the tree following the paths left or right based on comparisons. Once you find the leaf node, you can remove it from the tree without affecting the structure of the remaining tree, ensuring that it remains a valid search tree.

Imagine a library where each book represents a node in a search tree. If you want to remove a book that is on the shelf and has no other books around it (leaf node), you can simply take it out and leave the other books untouched. This demonstrates how easy it is to remove a leaf node without disturbing the rest of the collection.

Sometimes a deleted node might have only one child. For instance, supposing we delete 74. So, we come down to 74, now we want to delete this node, but it has the right child, but now what we can do, you can promote this child.

When deleting a node that has only one child, you can simplify the process by connecting the parent of the node to be deleted directly to its child. Continuing with the example of deleting node 74, if node 74 has a right child, you simply 'promote' that child to take the place of node 74 in the tree. This involves redirecting the parent pointer to the child node, effectively bypassing the deleted node and maintaining the search tree's structure.

Think of a family tree. If a parent (node) passes away (is deleted) but has one child (one child node), instead of leaving a gap, the family would simply connect the grandparents (the parent's parents) directly to the child, ensuring that the family lineage remains intact.

Now, what if the child to delete has two children? Suppose you want to delete 37. We identify 37 now 37 must go to at we cannot arbitrarily re structure the tree. So, one strategy that works is to make a whole there to remove the 37 and replace it by either its predecessor or successor.

When a node to be deleted has two children, we cannot simply remove it without adjusting the tree structure. Instead, we find a replacement for the deleted node. One common approach is to find the predecessor (the largest value in the left subtree) of the node being deleted. Once identified, we replace the value of the node to be deleted with that of the predecessor, ensuring that all values to the left remain smaller, and all values to the right remain larger. After this, we delete the predecessor node, which will be easier to remove since it will have at most one child.

Consider a situation where you're swapping out a player on a sports team (the node). If the player you're removing (node) is a key player with many teammates (two children), you can't just remove him. Instead, you bring in the player who has been performing well from the bench (the predecessor) to take their place. After the swap, you may have to let the player from the bench go if he is replaced (deleting the predecessor), ensuring the team remains cohesive.

All these operations that we have to describe now walk down a single path. Therefore, the worst-case complexity for any one of these operations is the height of the current given tree. If we have maintained some kind of a balanced tree, then the height is logarithmic.

Understanding the deletion process also involves recognizing that it typically follows a single path down the tree, resulting in a worst-case time complexity relative to the height of the tree. In well-maintained balanced trees, operations like insertion and deletion become very efficient, operating at a logarithmic time complexity due to the balance keeping tree height minimal.

Imagine a well-organized filing cabinet (the balanced tree) where documents are stored in an orderly fashion. When you look for a specific document or decide to remove one, you only need to navigate through a small number of sections (the height of the tree), making the entire process quick and efficient.

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