18.1. Case When Slope is -1

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 define a scenario where the slope is -1, meaning that the right subtree is taller than the left subtree. This situation arises when we are looking at balancing trees after certain operations, and the height of the balance, denoted by 'h', gives us context for how the tree looks. Here, 'h + 2' represents a possible increase in height due to nodes. It signals that we may need to perform certain rotations to maintain balance.

Consider a scale that is tipping to one side, specifically the right side. If we place a heavier weight on the right while keeping the left side lighter, the scale would tilt to the right. Similarly, in tree structures, if the right subtree grows much taller, the balance of our 'scale' (the tree) needs to be adjusted.

Now, this h plus 2 it now beings split of y and the rest therefore h plus 1 must be coming in the right and h 1 the left we got the slope by assumption we are assuming slope is minus 1. So, now we are going to expand out this node now it is a h plus 1, so there is at least 1 node here even a way which is 0 it has at least a root node. So, we will expose a root node as we exposed y earlier.

Continuing from our previous understanding, we take the identified node that represents 'h + 2' and split it. This yields a new height of 'h + 1' for the right and 'h' for the left. This process effectively makes the right subtree taller than its sibling, maintaining the assumption of 'minus 1' slope. The root node, therefore, emerges from this process, emphasizing that there’s a least one node present, establishing the tree's structure.

Think of building a tower with blocks. If you suddenly add extra blocks on the right side, that side becomes taller, tipping the tower. In evaluating which block to make the next support (new root), you expose the taller side while still maintaining some structure on the left.

So, now we are kind of expend it this TLR as z with 2 sub trees TLRL. So, this supposed to indicate let us started their original tree go and left go right and go left, so TLRL go left, go right, go right. So, it is TLRR, so that is notation for the sub trees, so this whole thing was h plus 1.

In this step, we explore subtrees represented as TLR (Tree Left Right) and denote them as separate structures under the tree. Notably, as we break down the tree into potential subtree configurations (like TLRL and TLRR), we can track the height adjustments we made previously. We see that the subtree configurations will help us in determining the heights and slopes as we prepare to adjust our tree back toward balance.

Picture a branching road system. As you're looking at which paths (subtrees) to take, recognizing that some lead left and others lead right helps in navigation. Similarly, breaking down complexities in tree structures into recognizable parts allows for better management of their heights and balance.

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.

Here, we outline a systematic approach for rebalancing the tree using a two-step rotation process. Specifically, we perform a rotation to the left to adjust node placement based on the identified configurations. This left rotation helps in redistributing the weight among the left and right subtrees to restore balance based on our set criteria.

Consider rearranging furniture in a room. If one side feels cluttered (too heavy), you might need to shift items around (rotate them) to create a more spacious feel. The same applies to tree structures — shifting nodes helps them feel balanced and organized.

So, to summarize this what we have said is that if we have plus 2, then we look at the left child of x call it y, if the slope of y is 0 or 1 we rotate at right at x, if the slope at y is minus 1 be first rotate this and then we rotate y.

To conclude our process, if we ascertain that a node has risen to a height of 'plus 2', we systematically assess its left child to determine the necessary rotations. Depending on the slope, different rotations need to be performed to maintain overall balance within the tree. We distinguish operations based on slope values, ensuring that all nodes return to acceptable height differentials.

Imagine your bookshelf where books grow taller on one side. If that side causes wobbling, you can either shift or bend a shelf to equalize everything. In trees, you assess the heights and make necessary movements to ensure all parts work in harmony.

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