Red-Black Trees | Top-Down Insertion

In Bottom-Up insertion of Red-Black Trees, “simple” Binary Search Tree insertion is used, followed by correction of the RB-Tree Violations on the way back up to the root. This can be done easily with the help of recursion. While in Top-Down Insertion, the corrections are done while traversing down the tree to the insertion point. When the actual insertion is done, no further corrections are needed, so no need to traverse back up the tree.

Therefore, the goal of Top-Down insertion is to traverse from the root to the insertion point in such a way that RB properties are maintained. This iterative approach thus makes Top-Down insertion faster than Bottom-Up insertion.

The two basic operations to perform for fixing violations and balancing are-



  • Recoloring
  • Rotation
  • Following is the detailed Algorithm
    The main goal of this algorithm is to create an insertion point at which the parent of the new node is Black, or the uncle of the new node is black.

    Let N be the new node to be inserted.

    1. If Y and Z are Black:
    2. perform Simple BST insertion. Insert new node N as the left/right child of Y OR Z and make the colour of the newly inserted node as red.

    3. If X’s Parent is Black:

    4. Then Recolor X, Y, Z and continue Down the tree.

    5. X’s Parent P is Red, Grandparent is Black and X and P are both left OR right children of Grandparent G:
      • Recolor X, Y, Z
      • Rotate P around G
      • Color P black
      • Color G red

      If violations exist continue with further cases.

    6. X’s Parent is Red, Grandparent is Black and X and P are opposite children of Grandparent G
      • Recolor X, Y, Z
      • Rotate X around P
      • Rotate X around G
      • Recolor X and G

      Insert New Node N at the required Position.

      Example :
      Insert Node 3 in the RB-Tree below –

    Below the implementation of the following approach:

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    // Java implementation for Top-Down
    // Red-Black Tree Insertion creating
    // a red black tree and storing an
    // English sentence into it using Top
    // down insertion approach
      
    import static java.lang.Integer.max;
      
    // Class for performing
    // RBTree operations
    public class RbTree {
        TreeNode Root = null;
      
        // Function to calculate
        // the height of the tree
        int HeightT(TreeNode Root)
        {
            int lefth, righth;
      
            if (Root == null
                || (Root.children == null
                    && Root.children[1] == null)) {
                return 0;
            }
            lefth = HeightT(Root.children[0]);
            righth = HeightT(Root.children[1]);
      
            return (max(lefth, righth) + 1);
        }
      
        // Function to check if
        // dir is equal to 0
        int check(int dir)
        {
            return dir == 0 ? 1 : 0;
        }
      
        // Function to check if a
        // node's color is red or not
        boolean isRed(TreeNode Node)
        {
            return Node != null
                && Node.color.equals("R");
        }
      
        // Function to perform
        // single rotation
        TreeNode SingleRotate(TreeNode Node,
                              int dir)
        {
            TreeNode temp
                = Node.children[check(dir)];
            Node.children[check(dir)]
                = temp.children[dir];
            temp.children[dir] = Node;
            Root.color = "R";
            temp.color = "B";
      
            return temp;
        }
      
        // Function to perform double rotation
        TreeNode DoubleRotate(TreeNode Node,
                              int dir)
        {
            Node.children[check(dir)]
                = SingleRotate(Node.children[check(dir)],
                               check(dir));
            return SingleRotate(Node, dir);
        }
      
        // Function to insert a new
        // node with given data
        TreeNode Insert(RbTree tree,
                        String data)
        {
            if (tree.Root == null) {
                tree.Root
                    = new TreeNode(data);
                if (tree.Root == null)
                    return null;
            }
            else {
      
                // A temporary root
                TreeNode temp = new TreeNode("");
      
                // Grandparent and Parent
                TreeNode g, t;
                TreeNode p, q;
      
                int dir = 0, last = 0;
      
                t = temp;
      
                g = p = null;
      
                t.children[1] = tree.Root;
      
                q = t.children[1];
                while (true) {
      
                    if (q == null) {
      
                        // Inserting root node
                        q = new TreeNode(data);
                        p.children[dir] = q;
                    }
      
                    // Sibling is red
                    else if (isRed(q.children[0])
                             && isRed(q.children[1])) {
      
                        // Recoloring if both
                        // children are red
                        q.color = "R";
                        q.children[0].color = "B";
                        q.children[1].color = "B";
                    }
      
                    if (isRed(q) && isRed(p)) {
      
                        // Resolving red-red
                        // violation
                        int dir2;
                        if (t.children[1] == g) {
                            dir2 = 1;
                        }
                        else {
                            dir2 = 0;
                        }
      
                        // If children and parent
                        // are left-left or
                        // right-right of grand-parent
                        if (q == p.children[last]) {
                            t.children[dir2]
                                = SingleRotate(g,
                                               last == 0
                                                   ? 1
                                                   : 0);
                        }
      
                        // If they are opposite
                        // childs i.e left-right
                        // or right-left
                        else {
                            t.children[dir2]
                                = DoubleRotate(g,
                                               last == 0
                                                   ? 1
                                                   : 0);
                        }
                    }
      
                    // Checking for correct
                    // position of node
                    if (q.data.equals(data)) {
                        break;
                    }
                    last = dir;
      
                    // Finding the path to
                    // traverse [Either left
                    // or right ]
                    dir = q.data.compareTo(data) < 0
                              ? 1
                              : 0;
      
                    if (g != null) {
                        t = g;
                    }
      
                    // Rearranging pointers
                    g = p;
                    p = q;
                    q = q.children[dir];
                }
      
                tree.Root = temp.children[1];
            }
      
            // Assign black color
            // to the root node
            tree.Root.color = "B";
      
            return tree.Root;
        }
      
        // Print nodes at each
        // level in level order
        // traversal
        void PrintLevel(TreeNode root, int i)
        {
            if (root == null) {
                return;
            }
      
            if (i == 1) {
                System.out.print("| "
                                 + root.data
                                 + " | "
                                 + root.color
                                 + " |");
      
                if (root.children[0] != null) {
                    System.out.print(" "
                                     + root.children[0].data
                                     + " |");
                }
                else {
                    System.out.print(" "
                                     + "NULL"
                                     + " |");
                }
                if (root.children[1] != null) {
                    System.out.print(" "
                                     + root.children[1].data
                                     + " |");
                }
                else {
                    System.out.print(" "
                                     + "NULL"
                                     + " |");
                }
      
                System.out.print(" ");
      
                return;
            }
      
            PrintLevel(root.children[0],
                       i - 1);
            PrintLevel(root.children[1],
                       i - 1);
        }
      
        // Utility Function to
        // perform level order
        // traversal
        void LevelOrder(TreeNode root)
        {
            int i;
      
            for (i = 1;
                 i < HeightT(root) + 1;
                 i++) {
                PrintLevel(root, i);
                System.out.print("\n\n");
            }
        }
    }
      
    // Class for representing
    // a node of the tree
    class TreeNode {
      
        // Class variables
        String data, color;
        TreeNode children[];
      
        public TreeNode(String data)
        {
            // Color R- Red
            // and B - Black
            this.data = data;
            this.color = "R";
            children
                = new TreeNode[2];
            children[0] = null;
            children[1] = null;
        }
    }
      
    // Driver Code
    class Driver {
        public static void main(String[] args)
        {
            // Tree Node Representation
            // -------------------------------------------
            // DATA | COLOR | LEFT CHILD | RIGHT CHILD |
            // -------------------------------------------
      
            RbTree Tree = new RbTree();
            String Sentence, Word;
            Sentence = "old is gold";
            String Word_Array[]
                = Sentence.split(" ");
      
            for (int i = 0;
                 i < Word_Array.length;
                 i++) {
                Tree.Root
                    = Tree.Insert(Tree,
                                  Word_Array[i]);
            }
      
            // Print Level Order Traversal
            System.out.println("The Level"
                               + "Order Traversal"
                               + "of the tree is:");
            Tree.LevelOrder(Tree.Root);
            System.out.println("\nInserting a"
                               + " word in the tree:");
            Word = "forever";
            Tree.Root = Tree.Insert(Tree,
                                    Word);
      
            System.out.println("");
            Tree.LevelOrder(Tree.Root);
        }
    }

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    Output:

    The LevelOrder Traversalof the tree is:
    | is | B | gold | old | 
    
    | gold | R | NULL | NULL | | old | R | NULL | NULL | 
    
    
    Inserting a word in the tree:
    
    | is | B | gold | old | 
    
    | gold | B | forever | NULL | | old | B | NULL | NULL | 
    
    | forever | R | NULL | NULL | 
    

    References:
    Red Black Trees – UMBC CSEE



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