 So I have made a few changes to our tree, really because I want to find this edge case where I happen to have a red sibling. I also did a quick change to one of the steps because now that I'm doing an assessment of it, I saw that I had a typo. But, let's say for example, I came in and I wanted to do a remove 75. So our rules would still apply. I find 75 and I would then find my in-order predecessor. In this case, it's 65. And as we've done in the past, I would take those and as you can see, I would spend some time erasing those numbers only to add 65 back to in. Now, the first step inside this, again, since we're dealing with a black node, X, I have to look at my sibling and this is one of the times where, again, I make an assessment. Well, it's not a black node, so I can't do the first two rules that I've done in the past. I would have to look at this now, red node. So in that case, the first thing I am asked to do is make X a double black node. This is allowing me to at least count things as a two when we're doing our black depth. However, since it's now in my tree, I'm actually out of balance. So I need to sort of resolve this. Now, you notice the next step is that I look at my parent and I look at my sibling and I'm going to perform a rotation on them. So we've already kind of discussed a rotation in the past. I have some Y and I have some X. They both have potential sub trees that they're both working off of. And so, you know, that T1, T2, T3. I take that rotation. In our case, I want to do a rotate left so that X becomes the new root of our little sub tree here. That is, Y is going to become the child of X. Y is going to get T1 and T2 while X gets T3. So in our case, I'll do it a little over here and we'll kind of expand on it later on. As we've noted, my X, which happens to be my S here. I know I'm using X over here as well. It's different than this one. My S would be my new root node. And then I would give in that 65 to be my new child node. Then I would inherit my T3. Well, T3 happens to be 95 in this situation while 65 gets 85 and my double black node. Now I'm going to kind of keep that in there for a second because one way you can think about what we're seeing here is reds want to fall. They want to fall as far as they can. And that's sort of where this is kind of coming in. So I want that red to drop down as best I can and I want that black to come upward. So I'm not there just yet because you can see that next statement is that I recolor S. Remember, this is still S. I recolor it black and I'm going to recolor the former parent, which happens to be in, to be red. So 85 is going to become black and that red I was talking about starts to move its way downward. And so 65 gets to become red. Now the issue is, just to kind of bring this down a little bit, the issue is I'm still left with a double black in my situation. And that's actually where you notice I happen to have this statement that I have to make another assessment on X. I might have to. This is actually one of those situations where I would have to. And since I happen to do this reassessment, I look at all the same problems as before. I say, well, what are my situations? Well, I happen to notice that I happen to have a sibling. I'll get rid of this S now. I happen to have a sibling, which is black and the children of that node are also black. Well, X has already been marked as double black, so we don't make it double, double black. We just leave it as is. However, what we're going to do in this situation, as you can see, is the next rule is I mark it as being a red node. I mark it sibling as being red. Again, you'll note that this is sort of how the color red that was originally here at 85 is moving downward. It moved to 65. Now, that's supposed to be an 81. Sorry. It's moving from 85 to 65 to 81. I once again still have the issue of 65 being red and having a red child, but that's where that last statement in my structure, in my previous instruction here, remove one black from X and push it to parent. So it's parent, that is. In this case, I would come in and I would strip that double black level off of X. And I would give it to 65. Since 65 was red, well, what do you happen to know? It gets recolored again. And so if I were to take this entire subtree that I've built out and bring it back to my entire tree, again, 25 is my root. It happens to have a 6 left node, which happens to have a 5 left node, which happens to have a 24 right node as well, a 21 left node. This is the side that effectively didn't get touched. However, when we start looking at the right side, we had a whole mess of things going on there. And so we see that I now have a black 85, a black 65, a black 95, and a red 81. I don't need to worry about X since it's a null node. I'm just kind of drawing it out. But if you wanted to, you could still envision them as being there for everything. Once again, we could double check that every rule is still being applied as well. So is my root black? Yes, it is. Are all of my children of red nodes black? Yes, they are. And what are the other rules? What's my black depth? Well, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4. So once again, I've now set up the rules. All right, I've done a proper red-black tree removal.