Deletion: Like the other basic operations on an n-node red-black tree, deletion of a node takes time O(lg n). Deleting a node from a red-black tree is only slightly more complicated than inserting a node.
The procedure RB-DELETE is a minor modification of the TREE-DELETE procedure. After splicing out a node, it calls an auxiliary procedure RB-DELETE-FIXUP that changes colors and performs rotations to restore the red-black properties.
RB-DELETE(T, z) 1 if left[z] = nil[T] or right[z] = nil[T] 2 then y ← z 3 else y ← TREE-SUCCESSOR(z) 4 if left[y] ≠ nil[T] 5 then x ← left[y] 6 else x ← right[y] 7 p[x] ← p[y] 8 if p[y] = nil[T] 9 then root[T] ← x 10 else if y = left[p[y]] 11 then left[p[y]] ← x 12 else right[p[y]] ← x 13 if y 3≠ z 14 then key[z] ← key[y] 15 copy y's satellite data into z 16 if color[y] = BLACK 17 then RB-DELETE-FIXUP(T, x) 18 return y
There are three differences between the procedures TREE-DELETE and RB-DELETE. First, all references to NIL in TREE-DELETE are replaced by references to the sentinel nil[T ] in RB-DELETE. Second, the test for whether x is NIL in line 7 of TREE-DELETE is removed, and the assignment p[x] ← p[y] is performed unconditionally in line 7 of RB-DELETE. Thus, if x is the sentinel nil[T], its parent pointer points to the parent of the spliced-out node y. Third, a call to RB-DELETE-FIXUP is made in lines 16-17 if y is black. If y is red, the red-black properties still hold when y is spliced out, for the following reasons:
no black-heights in the tree have changed,
no red nodes have been made adjacent, and
since y could not have been the root if it was red, the root remains black.
The node x passed to RB-DELETE-FIXUP is one of two nodes: either the node that was y's sole child before y was spliced out if y had a child that was not the sentinel nil[T ], or, if y had no children, x is the sentinel nil[T ]. In the latter case, the unconditional assignment in line 7 guarantees that x's parent is now the node that was previously y's parent, whether x is a key-bearing internal node or the sentinel nil[T].
We can now examine how the procedure RB-DELETE-FIXUP restores the red-black properties to the search tree.
RB-DELETE-FIXUP(T, x) 1 while x ≠ root[T] and color[x] = BLACK 2 do if x = left[p[x]] 3 then w ← right[p[x]] 4 if color[w] = RED 5 then color[w] ← BLACK ▹ Case 1 6 color[p[x]] ← RED ▹ Case 1 7 LEFT-ROTATE(T, p[x]) ▹ Case 1 8 w ← right[p[x]] ▹ Case 1 9 if color[left[w]] = BLACK and color[right[w]] = BLACK 10 then color[w] ← RED ▹ Case 2 11 x p[x] ▹ Case 2 12 else if color[right[w]] = BLACK 13 then color[left[w]] ← BLACK ▹ Case 3 14 color[w] ← RED ▹ Case 3 15 RIGHT-ROTATE(T, w) ▹ Case 3 16 w ← right[p[x]] ▹ Case 3 17 color[w] ← color[p[x]] ▹ Case 4 18 color[p[x]] ← BLACK ▹ Case 4 19 color[right[w]] ← BLACK ▹ Case 4 20 LEFT-ROTATE(T, p[x]) ▹ Case 4 21 x ← root[T] ▹ Case 4 22 else (same as then clause with "right" and "left" exchanged) 23 color[x] ← BLACK
If the spliced-out node y in RB-DELETE is black, three problems may arise. First, if y had been the root and a red child of y becomes the new root, we have violated property 2. Second, if both x and p[y] (which is now also p[x]) were red, then we have violated property 4. Third, y's removal causes any path that previously contained y to have one fewer black node. Thus, property 5 is now violated by any ancestor of y in the tree. We can correct this problem by saying that node x has an "extra" black. That is, if we add 1 to the count of black nodes on any path that contains x, then under this interpretation, property 5 holds. When we splice out the black node y, we "push" its blackness onto its child. The problem is that now node x is neither red nor black, thereby violating property 1. Instead, node x is either "doubly black" or "red-and-black," and it contributes either 2 or 1, respectively, to the count of black nodes on paths containing x. The color attribute of x will still be either RED (if x is red-and-black) or BLACK (if x is doubly black). In other words, the extra black on a node is reflected in x's pointing to the node rather than in the color attribute.
