Following article is extension of article discussed here.
In AVL tree insertion, we used rotation as a tool to do balancing after insertion caused imbalance. In Red-Black tree, we use two tools to do balancing.
1) Recoloring
2) Rotation
We try recoloring first, if recoloring doesn’t work, then we go for rotation. Following is detailed algorithm. The algorithms has mainly two cases depending upon the color of uncle. If uncle is red, we do recoloring. If uncle is black, we do rotations and/or recoloring.
Color of a NULL node is considered as BLACK.
Let x be the newly inserted node.
1) Perform standard BST insertion and make the color of newly inserted nodes as RED.
2) If x is root, change color of x as BLACK (Black height of complete tree increases by 1).
3) Do following if color of x’s parent is not BLACK or x is not root.
….a) If x’s uncle is RED (Grand parent must have been black from property 4)
……..(i) Change color of parent and uncle as BLACK.
……..(ii) color of grand parent as RED.
……..(iii) Change x = x’s grandparent, repeat steps 2 and 3 for new x.
….b) If x’s uncle is BLACK, then there can be four configurations for x, x’s parent (p) and x’s grandparent (g) (This is similar to AVL Tree)
……..i) Left Left Case (p is left child of g and x is left child of p)
……..ii) Left Right Case (p is left child of g and x is right child of p)
……..iii) Right Right Case (Mirror of case a)
……..iv) Right Left Case (Mirror of case c)
Following are operations to be performed in four subcases when uncle is BLACK.
All four cases when Uncle is BLACK
Left Left Case (See g, p and x)
Left Right Case (See g, p and x)
Right Right Case (See g, p and x)
Right Left Case (See g, p and x)
Examples of Insertion
Below is C++ Code.
/** C++ implementation for Red-Black Tree Insertion This code is adopted from the code provided by Dinesh Khandelwal in comments **/#include <bits/stdc++.h> using namespace std; enum Color {RED, BLACK}; struct Node { int data; bool color; Node *left, *right, *parent; // Constructor Node(int data) { this->data = data; left = right = parent = NULL; } }; // Class to represent Red-Black Tree class RBTree { private: Node *root; protected: void rotateLeft(Node *&, Node *&); void rotateRight(Node *&, Node *&); void fixViolation(Node *&, Node *&); public: // Constructor RBTree() { root = NULL; } void insert(const int &n); void inorder(); void levelOrder(); }; // A recursive function to do level order traversal void inorderHelper(Node *root) { if (root == NULL) return; inorderHelper(root->left); cout << root->data << " "; inorderHelper(root->right); } /* A utility function to insert a new node with given key in BST */Node* BSTInsert(Node* root, Node *pt) { /* If the tree is empty, return a new node */ if (root == NULL) return pt; /* Otherwise, recur down the tree */ if (pt->data < root->data) { root->left = BSTInsert(root->left, pt); root->left->parent = root; } else if (pt->data > root->data) { root->right = BSTInsert(root->right, pt); root->right->parent = root; } /* return the (unchanged) node pointer */ return root; } // Utility function to do level order traversal void levelOrderHelper(Node *root) { if (root == NULL) return; std::queue<Node *> q; q.push(root); while (!q.empty()) { Node *temp = q.front(); cout << temp->data << " "; q.pop(); if (temp->left != NULL) q.push(temp->left); if (temp->right != NULL) q.push(temp->right); } } void RBTree::rotateLeft(Node *&root, Node *&pt) { Node *pt_right = pt->right; pt->right = pt_right->left; if (pt->right != NULL) pt->right->parent = pt; pt_right->parent = pt->parent; if (pt->parent == NULL) root = pt_right; else if (pt == pt->parent->left) pt->parent->left = pt_right; else pt->parent->right = pt_right; pt_right->left = pt; pt->parent = pt_right; } void RBTree::rotateRight(Node *&root, Node *&pt) { Node *pt_left = pt->left; pt->left = pt_left->right; if (pt->left != NULL) pt->left->parent = pt; pt_left->parent = pt->parent; if (pt->parent == NULL) root = pt_left; else if (pt == pt->parent->left) pt->parent->left = pt_left; else pt->parent->right = pt_left; pt_left->right = pt; pt->parent = pt_left; } // This function fixes violations caused by BST insertion void RBTree::fixViolation(Node *&root, Node *&pt) { Node *parent_pt = NULL; Node *grand_parent_pt = NULL; while ((pt != root) && (pt->color != BLACK) && (pt->parent->color == RED)) { parent_pt = pt->parent; grand_parent_pt = pt->parent->parent; /* Case : A Parent of pt is left child of Grand-parent of pt */ if (parent_pt == grand_parent_pt->left) { Node *uncle_pt = grand_parent_pt->right; /* Case : 1 The uncle of pt is also red Only Recoloring required */ if (uncle_pt != NULL && uncle_pt->color == RED) { grand_parent_pt->color = RED; parent_pt->color = BLACK; uncle_pt->color = BLACK; pt = grand_parent_pt; } else { /* Case : 2 pt is right child of its parent Left-rotation required */ if (pt == parent_pt->right) { rotateLeft(root, parent_pt); pt = parent_pt; parent_pt = pt->parent; } /* Case : 3 pt is left child of its parent Right-rotation required */ rotateRight(root, grand_parent_pt); swap(parent_pt->color, grand_parent_pt->color); pt = parent_pt; } } /* Case : B Parent of pt is right child of Grand-parent of pt */ else { Node *uncle_pt = grand_parent_pt->left; /* Case : 1 The uncle of pt is also red Only Recoloring required */ if ((uncle_pt != NULL) && (uncle_pt->color == RED)) { grand_parent_pt->color = RED; parent_pt->color = BLACK; uncle_pt->color = BLACK; pt = grand_parent_pt; } else { /* Case : 2 pt is left child of its parent Right-rotation required */ if (pt == parent_pt->left) { rotateRight(root, parent_pt); pt = parent_pt; parent_pt = pt->parent; } /* Case : 3 pt is right child of its parent Left-rotation required */ rotateLeft(root, grand_parent_pt); swap(parent_pt->color, grand_parent_pt->color); pt = parent_pt; } } } root->color = BLACK; } // Function to insert a new node with given data void RBTree::insert(const int &data) { Node *pt = new Node(data); // Do a normal BST insert root = BSTInsert(root, pt); // fix Red Black Tree violations fixViolation(root, pt); } // Function to do inorder and level order traversals void RBTree::inorder() { inorderHelper(root);} void RBTree::levelOrder() { levelOrderHelper(root); } // Driver Code int main() { RBTree tree; tree.insert(7); tree.insert(6); tree.insert(5); tree.insert(4); tree.insert(3); tree.insert(2); tree.insert(1); cout << "Inoder Traversal of Created Tree
"; tree.inorder(); cout << "
Level Order Traversal of Created Tree
"; tree.levelOrder(); return 0; } |
Output:
Inoder Traversal of Created Tree 1 2 3 4 5 6 7 Level Order Traversal of Created Tree 6 4 7 2 5 1 3
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