17.1.6. Case Analysis for Rebalancing

Notion of balance that we will use is not with respect to size, but with respect to height. So, we will say that the height is a number of nodes from the root to a leaf. For example, if I have a tree which looks like this, then here the height is 1, 2, 3, 4 because on this path we have 4 nodes. So, the heights become 4 it is a length of the longest path measured in terms of nodes, the reason we measured in terms of nodes is, then we can distinguish easily, the empty tree from the tree with only are root.

This chunk introduces the concept of tree balance, emphasizing the difference between balance related to size and balance related to height. Height is defined as the count of nodes from the root to a leaf. The chunk also explains the practicality of measuring height in terms of nodes rather than edges, which helps distinguish between an empty tree and a tree with just the root.

Think of a tree as a staircase. The height represents the number of steps from the ground (root) to the top (leaf). Just like you can easily tell if there are steps present versus none, measuring tree height helps us easily identify if it is empty or has a single step (node).

And now in keeping with our earlier relaxation of the size condition, the height balance tree will be one where the height of the left and the height of the right differ the at most 1. Now, this is more relaxed in the previous thing.

A height-balanced tree is defined by the condition that the heights of the left and right subtrees can differ by at most 1. This relaxes the strict requirement of size balance, allowing for more flexibility. This definition is crucial for ensuring that the tree remains balanced during operations like insertion and deletion.

Imagine a seesaw where one side can be slightly lower or higher but should not differ too much. If one side goes too high compared to the other (more than one unit), the seesaw becomes unbalanced. Similarly, in a height-balanced tree, we aim for the left and right heights to remain close.

So, what we will end up to do is that whenever we do an insert or a delete we will immediately try to rebalance the tree. So, we would have a single disturbance from minus 2 or plus 2 it will never become very badly unbalance and we will immediately restore the balance to within minus 1 to plus 1.

This chunk explains the strategy of rebalancing the tree immediately after an insertion or deletion operation. It highlights that these operations can disturb the balance, creating a height difference of up to plus or minus 2. The objective is to restore balance to a range of minus 1 to plus 1 to maintain efficiency in operations.

Consider a tightrope walker who needs to quickly adjust their weight to maintain balance after a sudden shift. Similarly, as changes happen within the tree (insertions/deletions), we need to 'rebalance' the tree immediately to ensure it remains efficient and functional.

So, here is a typical situation that we would reach after a single operation which removes the balance. So, we might have a node which has slope plus 2 or minus 2, so let us look at plus 2 minus 2 turn out be symmetric.

This chunk discusses the scenario that arises after an operation (insert or delete) that causes the tree to become unbalanced, with a slope of plus or minus 2. Understanding the nature of this imbalance is critical, as it dictates the rebalancing strategy that will be employed to restore equilibrium.

Imagine a balancing scale that tips too far to one side after adding or removing weight. If this happens, the scale must be adjusted back to neutral. In tree operations, we analyze the nature of the imbalance (the slope) to determine how best to restore balance.

So, now since the left has height h plus 2, it has height at least 2, h can be at most at least smallest h can be 0, so it has height at least 2, so there at least 1 node here.

This chunk introduces the concept of rotations as a method for rebalancing the tree. When the tree is found to be heavily unbalanced (e.g., with a slope of plus 2), rotations are performed to rearrange the nodes, ensuring that the tree maintains balance. The operation hinges on understanding the heights of the subtrees involved.

Think of a carousel that has become tilted due to uneven weight distribution. To fix this, you might shift some riders (nodes) from one side to the other. Similarly, rotations allow us to redistribute nodes within a tree to achieve a balanced structure.

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