Python Data Structure and Algorithm Tutorial – Insertion in a Red-Black Tree

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Insertion in a Red-Black Tree

 

In this tutorial, you will learn how a new node can be inserted into a red-black tree is. Also, you will find working examples of insertions performed on a red-black tree in C, C++, Java and Python.

Red-Black tree is a self-balancing binary search tree in which each node contains an extra bit for denoting the color of the node, either red or black.

Before reading this article, please refer to the article on red-black tree.

While inserting a new node, the new node is always inserted as a RED node. After insertion of a new node, if the tree is violating the properties of the red-black tree then, we do the following operations.

  1. Recolor
  2. Rotation

 


Algorithm to Insert a New Node

Following steps are followed for inserting a new element into a red-black tree:

  1. The newNode be:
    New Node
    New node
  2. Let y be the leaf (ie. NIL) and x be the root of the tree. The new node is inserted in the following tree.
    insertion in red black tree
    Initial tree
  3. Check if the tree is empty (ie. whether x is NIL). If yes, insert newNode as a root node and color it black.
  4. Else, repeat steps following steps until leaf (NIL) is reached.
    1. Compare newKey with rootKey.
    2. If newKey is greater than rootKey, traverse through the right subtree.
    3. Else traverse through the left subtree.
      insertion in red black tree
      Path leading to the node where newNode is to be inserted
  5. Assign the parent of the leaf as parent of newNode.
  6. If leafKey is greater than newKey, make newNode as rightChild.
  7. Else, make newNode as leftChild.
    insertion in red black tree
    New node inserted
  8. Assign NULL to the left and rightChild of newNode.
  9. Assign RED color to newNode.
    insertion in red black tree
    Set the color of the newNode red and assign null to the children
  10. Call InsertFix-algorithm to maintain the property of red-black tree if violated.

Why newly inserted nodes are always red in a red-black tree?

This is because inserting a red node does not violate the depth property of a red-black tree.

If you attach a red node to a red node, then the rule is violated but it is easier to fix this problem than the problem introduced by violating the depth property.


Algorithm to Maintain Red-Black Property After Insertion

This algorithm is used for maintaining the property of a red-black tree if insertion of a newNode violates this property.

  1. Do the following until the parent of newNode p is RED.
  2. If p is the left child of grandParent gP of newNode, do the following.
    Case-I:

    1. If the color of the right child of gP of newNode is RED, set the color of both the children of gP as BLACK and the color of gP as RED.
      insertion in a red-black tree
      Color change
    2. Assign gP to newNode.
      insertion in a red-black tree
      Reassigning newNode

      Case-II:

    3. (Before moving on to this step, while loop is checked. If conditions are not satisfied, it the loop is broken.)
      Else if newNode is the right child of p then, assign p to newNode.

      insertion in a red-black tree
      Assigning parent of newNode as newNode
    4. Left-Rotate newNode.
      Insertion in a red-black tree
      Left Rotate

      Case-III:

    5. (Before moving on to this step, while loop is checked. If conditions are not satisfied, it the loop is broken.)
      Set color of p as BLACK and color of gP as RED.

      insertion in a redblack tree
      Color change
    6. Right-Rotate gP.
      insertion in a red-black tree
      Right Rotate
  3. Else, do the following.
    1. If the color of the left child of gP of z is RED, set the color of both the children of gP as BLACK and the color of gP as RED.
    2. Assign gP to newNode.
    3. Else if newNode is the left child of p then, assign p to newNode and Right-Rotate newNode.
    4. Set color of p as BLACK and color of gP as RED.
    5. Left-Rotate gP.
  4. (This step is performed after coming out of the while loop.)
    Set the root of the tree as BLACK.

    insertion in a red-black tree
    Set root’s color black

The final tree look like this:

insertion in a red-black tree
Final tree

Python Examples

/* Implementing Red-Black Tree in Python */


import sys

/* Node creation */
class Node():
    def __init__(self, item):
        self.item = item
        self.parent = None
        self.left = None
        self.right = None
        self.color = 1


class RedBlackTree():
    def __init__(self):
        self.TNULL = Node(0)
        self.TNULL.color = 0
        self.TNULL.left = None
        self.TNULL.right = None
        self.root = self.TNULL

    /* Preorder */
    def pre_order_helper(self, node):
        if node != TNULL:
            sys.stdout.write(node.item + " ")
            self.pre_order_helper(node.left)
            self.pre_order_helper(node.right)

    /* Inorder */
    def in_order_helper(self, node):
        if node != TNULL:
            self.in_order_helper(node.left)
            sys.stdout.write(node.item + " ")
            self.in_order_helper(node.right)

    /* Postorder */
    def post_order_helper(self, node):
        if node != TNULL:
            self.post_order_helper(node.left)
            self.post_order_helper(node.right)
            sys.stdout.write(node.item + " ")

    /* Search the tree */
    def search_tree_helper(self, node, key):
        if node == TNULL or key == node.item:
            return node

        if key < node.item:
            return self.search_tree_helper(node.left, key)
        return self.search_tree_helper(node.right, key)

    /* Balance the tree after insertion */
    def fix_insert(self, k):
        while k.parent.color == 1:
            if k.parent == k.parent.parent.right:
                u = k.parent.parent.left
                if u.color == 1:
                    u.color = 0
                    k.parent.color = 0
                    k.parent.parent.color = 1
                    k = k.parent.parent
                else:
                    if k == k.parent.left:
                        k = k.parent
                        self.right_rotate(k)
                    k.parent.color = 0
                    k.parent.parent.color = 1
                    self.left_rotate(k.parent.parent)
            else:
                u = k.parent.parent.right

                if u.color == 1:
                    u.color = 0
                    k.parent.color = 0
                    k.parent.parent.color = 1
                    k = k.parent.parent
                else:
                    if k == k.parent.right:
                        k = k.parent
                        self.left_rotate(k)
                    k.parent.color = 0
                    k.parent.parent.color = 1
                    self.right_rotate(k.parent.parent)
            if k == self.root:
                break
        self.root.color = 0

    /* Printing the tree */
    def __print_helper(self, node, indent, last):
        if node != self.TNULL:
            sys.stdout.write(indent)
            if last:
                sys.stdout.write("R----")
                indent += "     "
            else:
                sys.stdout.write("L----")
                indent += "|    "

            s_color = "RED" if node.color == 1 else "BLACK"
            print(str(node.item) + "(" + s_color + ")")
            self.__print_helper(node.left, indent, False)
            self.__print_helper(node.right, indent, True)

    def preorder(self):
        self.pre_order_helper(self.root)

    def inorder(self):
        self.in_order_helper(self.root)

    def postorder(self):
        self.post_order_helper(self.root)

    def searchTree(self, k):
        return self.search_tree_helper(self.root, k)

    def minimum(self, node):
        while node.left != self.TNULL:
            node = node.left
        return node

    def maximum(self, node):
        while node.right != self.TNULL:
            node = node.right
        return node

    def successor(self, x):
        if x.right != self.TNULL:
            return self.minimum(x.right)

        y = x.parent
        while y != self.TNULL and x == y.right:
            x = y
            y = y.parent
        return y

    def predecessor(self,  x):
        if (x.left != self.TNULL):
            return self.maximum(x.left)

        y = x.parent
        while y != self.TNULL and x == y.left:
            x = y
            y = y.parent

        return y

    def left_rotate(self, x):
        y = x.right
        x.right = y.left
        if y.left != self.TNULL:
            y.left.parent = x

        y.parent = x.parent
        if x.parent == None:
            self.root = y
        elif x == x.parent.left:
            x.parent.left = y
        else:
            x.parent.right = y
        y.left = x
        x.parent = y

    def right_rotate(self, x):
        y = x.left
        x.left = y.right
        if y.right != self.TNULL:
            y.right.parent = x

        y.parent = x.parent
        if x.parent == None:
            self.root = y
        elif x == x.parent.right:
            x.parent.right = y
        else:
            x.parent.left = y
        y.right = x
        x.parent = y

    def insert(self, key):
        node = Node(key)
        node.parent = None
        node.item = key
        node.left = self.TNULL
        node.right = self.TNULL
        node.color = 1

        y = None
        x = self.root

        while x != self.TNULL:
            y = x
            if node.item < x.item:
                x = x.left
            else:
                x = x.right

        node.parent = y
        if y == None:
            self.root = node
        elif node.item < y.item:
            y.left = node
        else:
            y.right = node

        if node.parent == None:
            node.color = 0
            return

        if node.parent.parent == None:
            return

        self.fix_insert(node)

    def get_root(self):
        return self.root

    def print_tree(self):
        self.__print_helper(self.root, "", True)


if __name__ == "__main__":
    bst = RedBlackTree()

    bst.insert(55)
    bst.insert(40)
    bst.insert(65)
    bst.insert(60)
    bst.insert(75)
    bst.insert(57)

    bst.print_tree()

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