 So one of the things I've gone ahead and done is we happen to have a lot of different cases to work off of, if X was red, if X was black, and you notice that's kind of a long list. Reason why is, let's say we wanted to be a little more adventurous. I came in and I said something like, remove 50. So we would still perform our normal removal process. As always, I would mark my 50 as my in, and I would need to find what we would classify as my X. Well, that's my in-order predecessor. So we would come down and we would note that 40 is my X. So what do we do? We take that X and we move that key up to our location. In our case, it happens to be root. So we go through that process. We do this before we make any assessments over here. That's the major thing that we want to focus in on. So I come in and I've removed all of those, and I now have 40 here. However, if I were to just kind of leave it as is, if I was finishing up here, my problem is if I were to kind of look at my black depth from all paths, especially my root, I would see that I have one black node, two black nodes, three black nodes, four black nodes, one black node, two black node, three black node. This, technically speaking, is a null node right now. It doesn't have children of itself. There are, so it would be stopping here. So what do we want to do? Well, this is actually kind of why, once again, I have these removal cases already up here. So if X happens to be black, I want to compare it to its sibling. So its sibling happens to be right beside it in this case. And we notice we have criteria for now, my sibling. In this case, we see that if the sibling is black, we check its children. Well, this child happens to be red. So you notice that first rule, if the sibling is black and both children are black as well, I would do some step. That's not the case. It's actually this second one right here. If sibling S is black and one of its children is red, we're going to perform a trinode restructuring. So for my sake, I'm going to call, I'm going to bring in the X, Ys and Zs again. I know this is going to be a little confusing because this happens to be an X. So for my sake, I'll call that R, X, Y and Z. So I still have that S. It's still, you know, it's still an S. It's still, we could call it R, we could call it X. It's kind of both of them. The big important thing is when I see this X here and this X here, they're not the same. They're two different kind of worlds. This one's being applied to sort of the removal process. This one's being applied to the trinode restructuring process. So we've seen that trinode restructuring a few times now and we still make the same assessments that we always make. We have to find out what is my A, what is my B and what is my C. Well, if I'm going from my left side first, we see that Y would actually be the first node because I would go to my left child. I see I have no left child here, so I then go to myself. Then the question is, well, where's next? This is all still happening on the left node, the left child of Z. So X would happen to be my B and that would leave me with my Y. Or sorry, my C as being Z. As always, as we've done in the past, B becomes the new parent, A and C are going to become the new children. And whatever gets inherited will be inherited. So I know there's another step. I'm going to skip over it for our sake just so we can continue kind of working through this for a second. So if I were to draw this out, I'll just kind of move down here a little bit so I don't need to actually even worry about root right now. I see that 24 would become the new X or the new B and then 25 would become the new C and then 21 would become the new A. Now, this still does not answer the question. You see that I have a red node with a red child and that's breaking the rules of our work. However, you see that next step. Notice how there's a next step as well. Color A and C, color A and C, black. And then give, this is kind of the important thing, give B whatever the former color of Z was. What is the former color of Z? Well, it's red, so we'd actually maintain that still redness but 25 in this case would be a black node. So just to kind of take this and look at it from our larger scale, if I were to come in here and I'm just going to do a little bit of deleting that 40 is from an olden times, as you can remember, and then I'd come in and I remove that as well. So once again, I would come in and now my parent node is 24 with two children, 21 and 25. Both of them are black. They happen to have their null children, which are classified as being black nodes as well. And if we were to make an assessment on everything, we'd see that I have one black node, two black node, three four, one, two, three, four, one, two, three, four. Every path happens to have four black nodes now.