1. Red-Black trees Red-black trees are 2-3-4 trees represented as binary trees at the expense of one extra bit per node. The idea is to represent 3- and 4-nodes as mini- binary trees, bound by “red” links. Red-black trees have the following properties: 13-nodes can be represented in two different ways. 2Two red links never follow one another in a row. 3All paths in the red-black tree have the same number of black links. 4One path containing red and black links can be twice as long as another path containing only black links. 5All path lengths are proportional to log N. 6Nodes with data items equal to a given node can fall on both sides of that node.

2. Examples 1 2 3 2 13 9 15 15 9 10 16 20 3 7 8 12 1318 915 25 16 1020 7 12 18 25 3 8 13 OR

3. Operations on red-black trees Search: same as in binary search trees, except for the need of a boolean attribute black that each node must maintain. Insert: same as in 2-3-4 trees, except that the colors of the links must be taken into consideration. Delete: similar to deletion in binary search trees. We can always delete a node that has at least one external child. However, if the key to be deleted is stored in a node with no external children, we move there the key of its inorder predecessor (or successor), and delete that node instead. Example: 7 5 4 8 4 8 2 5 9 2 9

4. Insert operation: example Insert 15 16 1020 12 18 25 13 7 3 8 16 1020 7 1318 25 3 8 15 Split 12

5. Splitting 4-nodes without rotation Case1: Splitting a 4-node connected to a 2-node

6. Splitting 4-nodes without rotation (contd.) Case2A: Splitting a 4-node connected to a 3-node

7. Splitting 4-nodes with single rotation Case2B: Splitting a 4-node connected to a 3-node

8. Splitting 4-nodes with double rotation Case2C: Splitting a 4-node connected to a 3-node Reduced to case 2B

9. The insertRB method Algorithm insertRB (T, newData, precedes) boolean success := false if (T = null) { NodeRBtree node = new NodeRBtree(newData) success := true } else { if (nodeType(T)= 4) splitroot(T) NodeRBtree p := T; NodeRBtree parent := null; boolean done := false while (! done) { if (nodeType(p) = 4) { if (nodeType (parent) = 2) then splitChildOf2 (p, parent) else splitChildOf3 (p, parent) } int compareResult := compare (p, newData, precedes) if (compareResult < 0) then success := false; done := true else if (compareResult = 0) then insertData (p, newData, precedes); success:=true; done:=true else advance (p, parent, compareResult) } // end while } // end else