04-12-2022, 07:51 PM
You pull the key out if it sits in a bottom leaf without trouble most times. But when the node drops below its minimum count you borrow a value from the side sibling to fill the gap. Or perhaps the sibling lacks enough so you combine the two nodes into one bigger piece. I notice this merge sends the problem up the chain because the parent loses a separator too. Then you repeat the fix higher until balance returns or the top root shrinks.
When the key rests inside a branch node you swap it first with the next value from the left subtree or right one. You delete that replacement instead since it lives at the bottom. And this swap keeps the order intact without breaking the search paths. But the underflow might still happen after so you check the counts again. Perhaps you end up borrowing or merging just like before but now the action ripples upward from there.
I watch you trace the whole path back up after each fix because changes affect the levels above. You verify the root stays with at least one key unless the tree shrinks to nothing. Or sometimes the root merges down and height drops by one level. But you avoid that by careful borrowing when possible to save height. And the whole process keeps the tree wide and shallow for fast lookups later on.
You deal with multiple keys in one node by shifting them around during merges or borrows. I think shifting moves values left or right to even out the load between spots. But uneven distribution can force extra merges up the line if not fixed promptly. Perhaps a single deletion cascades through several levels if the tree stays skinny. And you always start from the bottom fix and work your way to the top root.
The tree stays balanced because merges and borrows happen only when needed and they preserve the height across all paths. You see this balance lets searches stay quick even after many removals. Or maybe you test it by removing keys one after another and watching the structure adjust. But repeated deletions in the same area often trigger merges that clean up empty space. And the result keeps every leaf at the same depth no matter what.
I find the process tricky when keys repeat or sit near the minimum count because choices multiply. You pick the borrow direction based on which side has extra room to spare. Or you merge when both sides sit too low to lend anything. Perhaps the parent node itself needs fixing next which repeats the cycle higher. But careful tracking of counts at each step stops the whole thing from falling apart.
You consider the case where the root loses its last key and the tree drops a level entirely. I see that happen only after the final merge at the top. Or sometimes a borrow from a sibling saves the root from shrinking. But you check the root specially because it can hold fewer keys than others. And this special rule lets the height decrease naturally when the data thins out.
The adjustments after deletion ensure no node goes empty while keeping the order of values sorted throughout. You move separators between nodes during a merge to connect the parts correctly. Perhaps an example with small numbers shows how a leaf merge pulls a parent key down. But that pull might empty the parent so the wave continues upward. And you repeat until every node meets its size rule again.
I notice deletions work recursively because each fix might create a new problem one level up. You handle the leaf case first then climb to fix parents as needed. Or the borrow might solve it locally without climbing further. Perhaps you avoid climbing by choosing the right sibling early. But the tree properties force you to maintain order and counts at every point.
You explore different deletion patterns like removing from the middle versus the edge to see varied effects. I think middle removals often swap with nearby leaves and cause more shifts. Or edge removals might trigger fewer merges if neighbors stay full. But repeated patterns teach you when borrowing works better than joining. And the overall shape stays efficient for future searches after all fixes settle.
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