18. AVL Tree Rotations

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.

In AVL trees, we monitor the balance through something called the slope, which is determined by comparing the heights of the left and right subtrees. A slope of -1 indicates that the right subtree is taller than the left subtree. This suggests that we might need to make adjustments (or rotations) to maintain the AVL property, which asserts that the heights of the two child subtrees of any node differ by at most one.

Think of balancing scales. If one side is significantly heavier than the other, you need to either add weight to the lighter side or remove weight from the heavier side to restore balance. In AVL trees, when the scales tip too far to one side (slope -1), we need to rotate nodes to restore balance.

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.

When we encounter a -1 slope, we need to investigate further into the structure of our tree. The hypothesis states that with a height of h+2, the subtrees can be balanced by appropriately assigning heights. If we denote the heights of trees in a certain notation, we find the need to reassign tree heights based on existing nodes to maintain balance.

Imagine a see-saw with one side slightly raised. To make it level again, you might add a small weight to the lighter side or adjust the positioning of where people are sitting. In this case, you are redistributing heights or weights among the trees to achieve a balanced structure.

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.

If we determine that a left rotation is necessary, we proceed to 'rotate' nodes in a particular way. Here, we will focus on the subtree's upper node being rotated left, which incorporates specific nodes as new roots and rearranges their child subtrees. This structure is built gradually to ensure that all conditions for tree balance and binary search rules are preserved.

Picture a revolving door at a shopping mall. When the door swings to one side (left rotation), the people behind it adjust to the new opening position effectively. Similarly, when we rotate a tree to the left, the nodes adjust their positions around the new root to ensure everything remains organized.

If you look at the slope of y, then it is either 0 or it could be plus 1 because this could be h minus 1. If you look at x it could be minus 1, because this could be h minus 1 this definitely h or we could be 0.

After completing the rotation, we check the slopes of the involved nodes again. It is crucial that, following all rotations, the subtrees’ heights are still in compliance with AVL definitions, maintaining each node's slope within the acceptable range (-1, 0, or 1). Even if slight fluctuations occur, they must never exceed these thresholds to maintain balance.

Visualize adjusting the levels in a music mixer. Each adjustment slightly changes the sound (the height), but it must remain within the defined range for optimal balance. If one sound becomes too loud (slope exceeds 1), you need to adjust other sounds (subtrees) to keep harmony.

Rebalancing is these two steps and rotation this is set up other steps, so basically rebalancing at a given node is a constant number operations at that node.

Rebalancing is key to maintaining the integrity of the AVL Tree. After inserting or deleting nodes, we regularly check and possibly modify the tree structure to ensure it remains balanced. The operations required for rebalancing (rotations) are constant in terms of time complexity, meaning they are done efficiently without considerable slowdown, even in larger trees.

Think of organizing books in a library. After adding new books, librarians periodically check the organization system to ensure balance (genres on shelves) is maintained. Similarly, AVL trees regularly adjust their nodes to ensure that all operations remain efficient and accurate.

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