18.6. Rebalancing During Deletions

So, the third situation is that it is not 0 or plus 1, but the slope at whiles minus 1. So, we have already dealt with the case where 0 or plus 1 is the possibility. So, if it is minus 1 this means that the right sub tree must be strictly taller than the next left sub tree. So, with the whole thing remember again is h plus 2 the high assumption, because the whole thing has plus 2, so this whole thing is h plus 2 and this thing is h.

In this section, we are looking at a scenario where the slope of the tree is -1. The slope is calculated as the height of the left subtree minus the height of the right subtree. If the slope is -1, it indicates that the right subtree is taller than the left subtree. We generally denote the heights of nodes in the tree, and here we have two heights: h + 2 for the entire structure and h for one subtree. Identifying these heights is crucial for ensuring the tree remains balanced.

Imagine a teeter-totter or seesaw. If one side rises significantly higher than the other (representing an unbalanced tree), it’s harder to use properly. Just like you need to adjust the weight or position on the seesaw to balance it out, in tree structures, we need to adjust heights to keep the tree balanced.

So, now I have a two-step procedure what I will do is, now I will do is similar rotation, but to the left involving these two nodes. So, I will hang up these sub trees here by z.

To rebalance the tree, we perform rotations. The first step involves a left rotation of the subtree. The goal of this rotation is to rearrange the nodes so that the tree becomes more balanced. By doing this, we ensure that the heights of the subtrees become more equitable, thus resolving the slope issue. It's important to connect the subtrees correctly to maintain the overall structure of the tree.

Think of adjusting the position of chairs in a party. If one area is too crowded compared to another (unbalanced), you might move some chairs around to distribute people more evenly throughout the room (rebalancing). This makes it easier for guests to move around, just as rebalancing makes it easier to navigate through a tree.

But, we can get around this by just keeping the current value of the height and updating it each then. So, we have additional things, so we have in our tree node, we currently had the value, parent, left and right. So, now we just add one more field, so now whenever we do any rebalancing, then the height of this tree may change the current node.

To efficiently manage the height and slope of nodes after modifications, we can store the current height of each node directly within the node itself. This means that instead of recalculating every time to determine the height after an operation, we simply update the height field. This allows for quicker access and updates, ensuring efficient rebalancing during inserts or deletions.

Think about maintaining a plant. Instead of measuring its height every time you want to see how much it has grown, you can mark its height on the wall next to it. Every time you water it (akin to performing an operation in the tree), you check the mark and update it if necessary. This way, you keep track of growth without constantly re-measuring from the ground up.

The same with the delete, where above we do a recursive delete we rebalance.

Similar to the insertion process, when we delete a node, we must also ensure that the tree remains balanced. We perform a rebalance operation after each deletion recursively. This ensures that despite removing nodes, the tree maintains its balanced properties. The need to rebalance after deletion is just as critical as it is after insertion, to keep all operations efficient.

Consider a library where you remove and add books frequently. If you don’t reorganize the shelves after removing a few books, some sections may become too crowded while others might have empty spaces. By regularly reorganizing (rebalancing), you ensure that the library remains easy to navigate, similar to keeping a tree balanced.

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