==Удаление==
f cient implementation of the delete operation is a challenge in many symbol-table implementa- tions, and red-black trees are no exception. Industrial-strength implementations run to over 100 lines of code, and text books generally describe the operation in terms of detailed case studies, eschewing full implementations. Guibas and Sedgewick presented a delete implementation in [7], but it is not fully speci ed and depends on a call-by-reference approach not commonly found in modern code. The most popular method in common use is based on a parent pointers (see [6]), which adds substantial overhead and does not reduce the number of cases to be handled.The code on the next page is a full implementation of <tex>delete()</tex> for LLRB 2-3 trees. It is based on the reverse of the approach used for insert in top-down 2-3-4 trees: we perform rotations and color ips on the way down the search path to ensure that the search does not end on a 2-node, so that we can just delete the node at the bottom. We use the method <tex>fixUp()</tex> to share the code for the color ip and rotations following therecursive calls in the <tex>insert()</tex> code.With <tex>fixUp()</tex>, we can leave right-leaning red links and unbalanced 4-nodes along the search path, securethat these conditions will be xed on the way up the tree. (The approach is also effective 2-3-4 trees, but requires an extra rotation when the right nodeoff the search path is a 4-node.)As a warmup, consider the delete-the-minimum operation, where the goal is to delete the bottom node on the left spine while maintaining balance. To do so, we maintain the invariant that the current node or its left child is red. We can do so by moving to the left unless the current node is red and its left child and left grandchild are both black. In that case, we can do a color ip, which restores the invariant but may introduce successive reds on the right. In that case, we can correct the condition with two rotations and a color ip. These operations are implemented in the <tex>moveRedLeft()</tex> method on the next page. With <tex>moveRedLeft()</tex>, the recursive implementation of <tex>deleteMin()</tex> above is straightforward.For general deletion, we also need <tex>moveRedRight()</tex>, which is similar, but simpler, and we need to rotate left-leaning red links to the right on the search path to maintain the invariant. If the node to be deleted is an internal node, we replace its key and value elds with those in the minimum node in its right subtree and then delete the minimum in the right subtree (or we could rearrange pointers to use the node instead of copying elds). The full implementation of <tex>delete()</tex> that dervies from this discussion is given on the facing page. It uses one-third to one-quarter the amount of code found in typical implementations. It has been demonstrated before [2, 11, 13] that maintaining a eld in each node containing its height can lead to code for delete that is similarly concise, but that extra space is a high price to pay in a practical implementation. With <tex>LLRB</tex> trees, we can arrange for concise code having a logarithmic performance guarantee and using no extra space.
