Figure 1949 A routine for performing an appropriate rotation

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Figure 1950 X has two black children, and both of its sibling's children are black; do a color flip

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lucky, we will fall onto a red node (we have at least a 50 percent chance that this will happen), thereby making the new current node red Otherwise, we have the situation shown in Figure 1953 That is, the current X is black, the current T is red, and the current P is black We can then rotate T and P, thereby making X's new parent red; X and its new grandparent are black Now X is not yet red, but we are back to the starting point (although one

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Figure 1951 X has two black children, and the outer child of its sibling is red; do a single rotation

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Figure 1952 X has two black children, and the inner child of its sibling is red; do a double rotation

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Figure 1953 Xis black, and at least one child is red; if we fall through to the next level and land on a red child, fine; if not, we rotate a sibling and parent

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level deeper) This outcome is good enough because it shows that we can iteratively descend the tree Thus, so long as we eventually either reach a node that has two black children or land on a red node, we are okay This result is guaranteed for the deletion algorithm because the two eventual states are

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X is a leaf, which is always handled by the main case since X has two black children; and X has only one child, for which the main case applies if the child is black, and if it is red, we can delete X, if necessary, and make the child black

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Lazy deletion, in which items are marked as deleted but not actually deleted, is sometimes used However, lazy deletion wastes space and complicates other routines (see Exercise 1926)

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Because of many possible rotations, the red-black tree is fairly tricky to code In particular, the remove operation is quite challenging In this section we describe a simple but competitive balanced search tree known as an AAtree The AA-tree is the method of choice when a balanced tree is needed, a casual implementation is acceptable, and deletions are needed The AA-tree adds one extra condition to the red-black tree: Left children may not be red This simple restriction greatly simplifies the red-black tree algorithms for two reasons: First, it eliminates about half of the restructuring cases; second, it simplifies the remove algorithm by removing an annoying case That is, if an internal node has only one child, the child must be a red right child because red left children are now illegal, whereas a single black child would violate property 4 for red-black trees Thus we can always replace an internal node with the smallest node in its right subtree That smallest node is either a leaf or has a red child and can be easily bypassed and removed To simplify the implementation further, we represent balance information in a more direct way Instead of storing a color with each node, we store the node's level The level of a node represents the number of left links on the path to the nu1 lNode sentinel and is level 1, if the node is a leaf; the level of its parent, if the node is red; and one less than the level of its parent, if the node is black The result is an AA-tree If we translate the structure requirement from colors to levels, we know that the left child must be one level lower than its parent and that the right child may be zero or one level lower than its parent (but not more) A horizontal link is a connection between a node and a child of equal levels The coloring properties imply that

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1 horizontal links are right links (because only right children may be red), 2 there may not be two consecutive horizontal links (because there cannot be consecutive red nodes), 3, nodes at level 2 or higher must have two children, and 4, if a node does not have a right horizontal link, its two children are at the same level

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The AA-tree is the method of choice when a balanced tree is needed, a casual implementation is acceptable, and deletions are needed

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The level of a node in an AA-tree represents the number of left links on the path to the nullNode sentinel

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A horizontal link in an AA-tree is a connection between a node and a child of equal levels A horizontal link should go only to the right, and there should not be two consecutive horizontal links

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