Red-Black Trees | Top-Down Insertion

Difficulty Level : Expert
Last Updated : 26 Dec, 2019

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.

If Y and Z are Black:
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.
If X’s Parent is Black:

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

Java

// 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);
    }
}

C#

// C# 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
using System;
  
// Class for performing
// RBTree operations
class RbTree
{
    public TreeNode Root = null;
  
    // Function to calculate
    // the height of the tree
    public 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 (Math.Max(lefth, righth) + 1);
    }
  
    // Function to check if
    // dir is equal to 0
    public int check(int dir)
    {
        return dir == 0 ? 1 : 0;
    }
  
    // Function to check if a
    // node's color is red or not
    public bool isRed(TreeNode Node)
    {
        return Node != null && 
               Node.color.Equals("R");
    }
  
    // Function to perform
    // single rotation
    public 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
    public 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
    public 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
    public void PrintLevel(TreeNode root, int i)
    {
        if (root == null)
        {
            return;
        }
  
        if (i == 1)
        {
            Console.Write("| " + root.data +
                         " | " + root.color + " |");
  
            if (root.children[0] != null) 
            {
                Console.Write(" " + 
                   root.children[0].data + " |");
            }
            else 
            {
                Console.Write(" " + "NULL" + " |");
            }
            if (root.children[1] != null)
            {
                Console.Write(" " + 
                   root.children[1].data + " |");
            }
            else 
            {
                Console.Write(" " + "NULL" + " |");
            }
  
            Console.Write(" ");
  
            return;
        }
  
        PrintLevel(root.children[0], i - 1);
        PrintLevel(root.children[1], i - 1);
    }
  
    // Utility Function to perform 
    // level order traversal
    public void LevelOrder(TreeNode root)
    {
        int i;
  
        for (i = 1; i < HeightT(root) + 1; i++)
        {
            PrintLevel(root, i);
            Console.Write("\n\n");
        }
    }
}
  
// Class for representing
// a node of the tree
public class TreeNode 
{
  
    // Class variables
    public String data, color;
    public 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
public 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";
        char[] spearator = { ' ', ' ' }; 
        String []Word_Array = Sentence.Split(spearator, 
                StringSplitOptions.RemoveEmptyEntries);
  
        for (int i = 0; i < Word_Array.Length; i++)
        {
            Tree.Root = Tree.Insert(Tree,
                            Word_Array[i]);
        }
  
        // Print Level Order Traversal
        Console.WriteLine("The Level" + 
                          "Order Traversal" + 
                          "of the tree is:");
        Tree.LevelOrder(Tree.Root);
        Console.WriteLine("\nInserting a" + 
                          " word in the tree:");
        Word = "forever";
        Tree.Root = Tree.Insert(Tree, Word);
  
        Console.WriteLine("");
        Tree.LevelOrder(Tree.Root);
    }
}
  
// This code is contributed by Rajput-Ji

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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