Add examples of red-black tree algorithms

+ Using pseudocode examples from MIT Introduction to Algorithms

cpp/algorithms/trees/redblack/driver.cpp

@@ -15,12 +15,15 @@
 
 int main (const int argc, const char * argv[])
 {
-  BinarySearchTree testTree;
+  RedBlackTree testTree;
-  std::vector<int> inputValues {2, 6, 9, 1, 5, 8, 10};
+//  std::vector<int> inputValues {2, 6, 9, 1, 5, 8, 10};
   // Alternative input values
-//  std::vector<int> inputValues {10, 12, 6, 4, 20, 8, 7, 15, 13};
+  // + Provides a larger RBT, where every path from the root to a leaf node
+  // ++ Also constructs the same RBT as seen in MIT Intro to Algorithms
+  std::vector<int> inputValues {26, 17, 41, 14, 21, 30, 47, 10, 16, 19, 23, 28,
+                                38, 7, 12, 15, 20, 35, 39, 3};
 
-  std::cout << "Building binary search tree with input: ";
+  std::cout << "Building red-black tree with input: ";
   for (const auto &value : inputValues) std::cout << value << ", ";
   std::cout << std::endl;
 
@@ -44,6 +47,7 @@ int main (const int argc, const char * argv[])
   // Test removing a node, printing the result in-order
   std::cout << "\nRemoving root value...\n";
   testTree.remove(testTree.getRoot()->element);
+  testTree.remove(testTree.getRoot()->element);
   // Can use inline function to remove a value directly for testing -
 //  testTree.remove(6);
 
@@ -53,20 +57,20 @@ int main (const int argc, const char * argv[])
 
   // Test copy constructor
   std::cout << "\nCloning testTree to testTree2...\n";
-  BinarySearchTree testTree2 = testTree;
+  RedBlackTree testTree2 = testTree;
   std::cout << "Inorder traversal of the cloned testTree2: \n";
   testTree2.printInOrder();
   std::cout << std::endl;
 
   // Test assignment operator
   std::cout << "\nCloning testTree to testTree3...\n";
-  BinarySearchTree testTree3;
+  RedBlackTree testTree3;
   testTree3 = testTree;
   std::cout << "Inorder traversal of the cloned testTree3 tree: \n";
   testTree3.printInOrder();
   std::cout << std::endl;
 
-  // Test emptying the BST
+  // Test emptying the RBT
   std::cout << "\nEmptying testTree...\n";
   testTree.makeEmpty();
   std::cout << "testTree isEmpty: " << (testTree.isEmpty() ? "true" : "false");
@@ -82,15 +86,15 @@ int main (const int argc, const char * argv[])
   testTree3.printInOrder();
   std::cout << std::endl;
 
-  std::cout << "\nChecking if tree contains value of 6: ";
+  std::cout << "\nChecking if tree contains value of 7: ";
-  std::cout << (testTree2.contains(6) ? "true" : "false") << std::endl;
+  std::cout << (testTree2.contains(7) ? "true" : "false") << std::endl;
   std::cout << "Checking if tree contains value of 600: ";
-  std::cout << (testTree2.contains(600) ? "true" : "false") << std::endl;
+  std::cout << (testTree2.contains(700) ? "true" : "false") << std::endl;
 
-  std::cout << "\nSuccessor of node with value 6: "
+  std::cout << "\nSuccessor of node with value 7: "
-            << testTree2.successor(testTree2.search(6))->element;
+            << testTree2.successor(testTree2.search(7))->element;
-  std::cout << "\nPredecessor of node with value 6: "
+  std::cout << "\nPredecessor of node with value 7: "
-            << testTree2.predecessor(testTree2.search(6))->element;
+            << testTree2.predecessor(testTree2.search(7))->element;
   std::cout << std::endl;
 
 }

cpp/algorithms/trees/redblack/redblack.cpp

@@ -15,76 +15,94 @@
 * Constructors, Destructors, Operators
 *******************************************************************************/
 
-/** BinarySearchTree Copy Assignment Operator
+// Use a static member for nil
- * @brief Empty the calling object's root BinaryNode, and copy the rhs data
+// + All RBTs will share this value to represent values Not In the List
+RedBlackTree::RedBlackNode *RedBlackTree::nil = new RedBlackTree::RedBlackNode();
+
+
+/*  RedBlackNode Copy Constructor
  *
- * Runs in O( n ) time, since we visit each node in the BST once
+ * Runs in O( n ) time, since we visit each node in the RBT once
- * + Where n is the total number of nodes within the BST
+ * + Where n is the total number of nodes within the RBT
+ *
+ * @param rhs An existing RBT to initialize this node (and children) with
+ */
+RedBlackTree::RedBlackNode::RedBlackNode(RedBlackNode * toCopy)
+{
+  // Base case, breaks recursion when we hit a null node
+  // + Returns to the previous call in the stack
+  if (toCopy == nil) return;
+  // Set the element of this RedBlackNode to the value in toCopy->element
+  element = toCopy->element;
+  // If there is a left / right node, copy it using recursion
+  //  + If there is no left / right node, set them to nullptr
+  if (toCopy->left != nil) {
+    left = new RedBlackNode(toCopy->left);
+    left->parent = this;
+  }
+  else left = nil;
+
+  if (toCopy->right != nil) {
+    right = new RedBlackNode(toCopy->right);
+    right->parent = this;
+  }
+  else right = nil;
+
+  if (toCopy->parent == nil) parent = nil;
+
+  // TODO: Fix the copying of the RBT
+  color = toCopy->color;
+}
+
+/** RedBlackTree Copy Assignment Operator
+ * @brief Empty the calling object's root RedBlackNode, and copy the rhs data
+ *
+ * Runs in O( n ) time, since we visit each node in the RBT once
+ * + Where n is the total number of nodes within the RBT
  *
  * makeEmpty() and clone() are both O( n ), and we call each sequentially
  * + This would appear to be O( 2n ), but we drop the constant of 2
  *
- * @param rhs The BST to copy, beginning from its root BinaryNode
+ * @param rhs The RBT to copy, beginning from its root RedBlackNode
- * @return BinarySearchTree The copied BinarySearchTree object
+ * @return RedBlackTree The copied RedBlackTree object
  */
-BinarySearchTree& BinarySearchTree::operator=(const BinarySearchTree &rhs)
+RedBlackTree& RedBlackTree::operator=(const RedBlackTree &rhs)
 {
   // If the objects are already equal, do nothing
   if (this == &rhs) return *this;
 
   // Empty this->root
   makeEmpty(root);
+
+//  if (root == nil) root = new RedBlackNode();
   // Copy rhs to this->root
   root = clone(rhs.root);
   return *this;
 }
 
-/*  BinaryNode Copy Constructor
+RedBlackTree::RedBlackTree(const RedBlackTree &rhs) {
- *
+  root = clone(rhs.getRoot());
- * Runs in O( n ) time, since we visit each node in the BST once
- * + Where n is the total number of nodes within the BST
- *
- * @param rhs An existing BST to initialize this node (and children) with
- */
-BinarySearchTree::BinaryNode::BinaryNode(BinaryNode * toCopy)
-{
-  // Base case, breaks recursion when we hit a null node
-  // + Returns to the previous call in the stack
-  if (toCopy == nullptr) return;
-  // Set the element of this BinaryNode to the value in toCopy->element
-  element = toCopy->element;
-  // If there is a left / right node, copy it using recursion
-  //  + If there is no left / right node, set them to nullptr
-  if (toCopy->left != nullptr) {
-    left = new BinaryNode(toCopy->left);
-    left->parent = this;
 }
-  if (toCopy->right != nullptr) {
-    right = new BinaryNode(toCopy->right);
-    right->parent = this;
-  }
-}
-
 
 /*******************************************************************************
 * Public Member Functions
 *******************************************************************************/
 
 /** contains
- * @brief Determines if value exists within a BinaryNode and its children
+ * @brief Determines if value exists within a RedBlackNode and its children
  *
- * Runs in O( height ) time, given the height of the current BST
+ * Runs in O( height ) time, given the height of the current RBT
- * + In the worst case, we search for a node at the bottom of the BST
+ * + In the worst case, we search for a node at the bottom of the RBT
  *
- * @param value The value to search for within the BST
+ * @param value The value to search for within the RBT
- * @param start The root BinaryNode to begin the search
+ * @param start The root RedBlackNode to begin the search
  * @return true If the value is found within the root node or its children
  * @return false If the value is not found within the root node or its children
  */
-bool BinarySearchTree::contains(const int &value, BinaryNode *start) const
+bool RedBlackTree::contains(const int &value, RedBlackNode *start) const
 {
   // If tree is empty
-  if (start == nullptr) return false;
+  if (start == nil) return false;
     // If x is smaller than our current value
   else if (value < start->element) return contains(value, start->left);
     // If x is larger than our current value, check the right node
@@ -92,59 +110,373 @@ bool BinarySearchTree::contains(const int &value, BinaryNode *start) const
   else return true;
 }
 
+/** rotateLeft
+ * @brief Rotates a node and it's children counter-clockwise around a pivotNode
+ *
+ * Runs in O( 1 ) constant time, only immediately available values are used
+ *
+ * @param pivotNode The node to begin rotation from
+ */
+void RedBlackTree::rotateLeft(RedBlackNode *pivotNode)
+{
+  // To rotateRight, we must relocate the rightChild node
+  RedBlackNode *rightChild = pivotNode->right;
+
+  pivotNode->right = rightChild->left;
+  // If the rightChild->left node exists, update it's parent to the pivotNode
+  if (rightChild->left != nil) rightChild->left->parent = pivotNode;
+  // After we rotateLeft, the rightChild will be in the position of pivotNode
+  // + So we update rightChild->parent to reflect this
+  rightChild->parent = pivotNode->parent;
+
+  // The following conditions relocate the rightChild to its new position
+  // Case 1: pivotNode has no parent, so it must be the rootNode
+  // Case 2: pivotNode is the left child of its parent node
+  // Case 2: pivotNode is the right child of its parent node
+  if (pivotNode->parent == nil) root = rightChild;
+  else if (pivotNode == pivotNode->parent->left) {
+    pivotNode->parent->left = rightChild;
+  }
+  else pivotNode->parent->right = rightChild;
+
+  // The rightChild is now the parent of pivotNode
+  // + Since we rotated left, set rightChild->left to point to the pivotNode
+  // + Update the pivotNode->parent to reflect this
+  rightChild->left = pivotNode;
+  pivotNode->parent = rightChild;
+}
+
+/** rotateRight
+ * @brief Rotates a node and it's children clockwise around a pivotNode
+ *
+ * Runs in O( 1 ) constant time, only immediately available values are used
+ *
+ * @param pivotNode The node to begin rotation from
+ */
+void RedBlackTree::rotateRight(RedBlackTree::RedBlackNode *pivotNode)
+{
+  // To rotateRight, we must relocate the leftChild node
+  RedBlackNode *leftChild = pivotNode->left;
+
+  pivotNode->left = leftChild->right;
+  // If the leftChild->left node exists, update it's parent to the pivotNode
+  if (leftChild->left != nil) leftChild->right->parent = pivotNode;
+  // After we rotateRight, the leftChild will be in the position of pivotNode
+  // + So we update leftChild->parent to reflect this
+  leftChild->parent = pivotNode->parent;
+
+  // The following conditions relocate the leftChild to its new position
+  // Case 1: pivotNode has no parent, so it must be the rootNode
+  // Case 2: pivotNode is the left child of its parent node
+  // Case 2: pivotNode is the right child of its parent node
+  if (pivotNode->parent == nil) root = leftChild;
+  else if (pivotNode == pivotNode->parent->left) {
+    pivotNode->parent->left = leftChild;
+  }
+  else pivotNode->parent->right = leftChild;
+
+  // The leftChild is now the parent of pivotNode
+  // + Since we rotated right, set leftChild->right to point to the pivotNode
+  // + Update the pivotNode->parent to reflect this
+  leftChild->right = pivotNode;
+  pivotNode->parent = leftChild;
+}
+
+/** insertFixup
+ * @brief Corrects RBT properties to enforce proper node colors after insertion
+ *
+ * Runs in O( lg(n) ) time, where n is the number of nodes in the RBT
+ *
+ * @param start The node to begin the fixup operation from
+ */
+void RedBlackTree::insertFixup(RedBlackTree::RedBlackNode *start)
+{
+  // While the parent of start node is valid and colored red
+  // + Check for the two red nodes in a row, and update their colors
+  while (start->parent != nil && start->parent->color == Red) {
+
+    // If the parent of start node is a *left* child
+    if (start->parent == start->parent->parent->left) {
+      // Check the *right* uncle node to enforce RBT properties
+      // + Since the start->parent->right would be the start node's sibling
+      // ++ The start->parent->parent->left would be the start node's uncle
+      RedBlackNode *uncleNode = start->parent->parent->right;
+
+      // If the uncleNode is red, we are in case 1
+      // + In case 1, start->parent->color and uncleNode->color are both Red
+      if (uncleNode->color == Red) {
+        start->parent->color = Black;
+        uncleNode->color = Black;
+        start->parent->parent->color = Red;
+        // Advance start to it's grandparent node
+        start = start->parent->parent;
+      }
+      else { // Otherwise, if case 1 is not violated...
+
+        // If the start node is a right child, we are in case 2
+        // + In case 2, uncleNode is Black, and start node is a right child
+        if (start == start->parent->right) {
+          // Advance start up the tree to it's parent node, then *rotateLeft*
+          start = start->parent;
+          rotateLeft(start);
+          // After this rotation, we have forced ourselves into case 3
+          // + So, whether this case (2) executes or not, we end up in case 3
+        }
+
+        // No need for an if statement here, since the start node must be either
+        // + a right or left child. It cannot be neither always run these steps.
+        // ++ The only time these steps are skipped, is when we are in case 1
+
+        // Start node is a left child, we are in case 3
+        // + In case 3, uncleNode is Black and start node is a left child
+        start->parent->color = Black;
+        start->parent->parent->color = Red;
+        // Rotate around the grandparent node
+        rotateRight(start->parent->parent);
+      }
+
+    }
+    else { // If the parent of start node is a *right* child
+      // We follow the same 3 cases as above..
+      // + but with all left / right rotations and references swapped
+
+      // Check the *left* uncle node to enforce RBT properties
+      RedBlackNode *uncleNode = start->parent->parent->left;
+      // If the uncleNode is red, we are in case 1
+      if (uncleNode != nil && uncleNode->color == Red) {
+        start->parent->color = Black;
+        uncleNode->color = Black;
+        start->parent->parent->color = Red;
+        // Advance start to it's grandparent node
+        start = start->parent->parent;
+      }
+      else { // Otherwise, if case 1 is not violated...
+
+        // If the start node is a left child, we are in case 2
+        if (start == start->parent->left) {
+          // Advance start up the tree to it's parent node, then *rotateRight*
+          start = start->parent;
+          rotateRight(start);
+          // After this rotation, we have forced ourselves into case 3
+          // + So, whether this case (2) executes or not, we end up in case 3
+        }
+
+        // The only time these steps are skipped, is when we are in case 1
+        // + Always perform these steps when not in case 1
+
+        // Start node is a *right* child, we are in case 3
+        start->parent->color = Black;
+        start->parent->parent->color = Red;
+        // Rotate around the grandparent node
+        rotateLeft(start->parent->parent);
+      }
+    }
+
+    // The while() loop will always terminate after case 2 and (or) 3 is ran
+  }
+
+  // The root of the RBT should always be black
+  root->color = Black;
+}
+
+/** deleteFixup
+ * @brief Corrects RBT properties to enforce proper node colors after removal
+ *
+ * Runs in O( lg(n) ) time, where n is the number of nodes in the RBT
+ *
+ * @param start The node to begin the fixup operation from
+ */
+void RedBlackTree::deleteFixup(RedBlackTree::RedBlackNode *start)
+{
+  // Until we reach the root of the RBT, move the extra black node up the tree
+  while (start != root && start->color == Black) {
+    // If the start node is a *left* child
+    if (start == start->parent->left) {
+
+      RedBlackNode *siblingNode = start->parent->right;
+      // If start node's siblingNode is colored Red, we are in case 1
+      // + In case 1, the only requirement is siblingNode is Red
+      if (siblingNode->color == Red) {
+        // Color the siblingNode Black, and the start->parent node Red
+        siblingNode->color = Black;
+        start->parent->color = Red;
+
+        // rotateLeft around the parent node, and update the siblingNode
+        // + siblingNode now represents the new sibling
+        // + start->parent is now a Red node with two Black siblings
+        // ++ This is ideal, since Red nodes can only have Black children nodes
+        rotateLeft(start->parent);
+        siblingNode = start->parent->right;
+      }
+
+      // If the siblingNode is Black, we are in case 2
+      // + In case 2, siblingNode is Black and has two Black children nodes
+      if (siblingNode->left->color == Black && siblingNode->right->color == Black) {
+        // Color the siblingNode Red and advance start node up the tree
+        siblingNode->color = Red;
+        start = start->parent;
+      }
+      else { // If either one of the siblingNode's children are Red...
+
+        // If the siblingNode->right child node is Black, we are in case 3
+        // + In case 3, the start node's siblingNode is black
+        // ++ And the siblingNode's right child is Black
+        if (siblingNode->right->color == Black) {
+          // Color the left child of siblingNode Black, and siblingNode to Red
+          siblingNode->left->color = Black;
+          siblingNode->color = Red;
+
+          // rotateRight(siblingNode) places the Red siblingNode as the right
+          // + child of the previous siblingNode->left
+          rotateRight(siblingNode);
+          // Update the siblingNode of start node to reflect the new sibling
+          siblingNode = start->parent->right;
+          // After rotateRight(siblingNode), we have put ourselves into case 4
+        }
+
+        // siblingNode->left is colored Black, we are in case 4
+        // + In case 4, the start node's sibling is Black
+        // ++ And the siblingNode's right child is Red
+        siblingNode->color = start->parent->color;
+        start->parent->color = Black;
+        siblingNode->right->color = Black;
+        // rotateLeft around the start->parent node, placing the siblingNode
+        // + In the previous position of the start->parent node
+        rotateLeft(start->parent);
+        // The previous start->parent node is now siblingNode's left child
+
+        // Setting the start node to root node ensures the while() terminates
+        start = root;
+      }
+    }
+    else { // If the start node is a right child
+      // We follow the same 3 cases as above..
+      // + but with all left / right rotations and references swapped
+
+      RedBlackNode *siblingNode = start->parent->left;
+      // If start node's siblingNode is colored Red, we are in case 1
+      if (siblingNode->color == Red) {
+        // Color the siblingNode Black, and the start->parent node Red
+        siblingNode->color = Black;
+        start->parent->color = Red;
+
+        // *rotateRight* around the parent node, and update the siblingNode
+        rotateRight(start->parent);
+        siblingNode = start->parent->left;
+      }
+
+      // If the siblingNode is Black, we are in case 2
+      if (siblingNode->left->color == Black && siblingNode->right->color == Black) {
+        siblingNode->color = Red;
+        start = start->parent;
+      }
+      else { // If either one of the siblingNode's children are Red...
+
+        // If the siblingNode->left child node is Black, we are in case 3
+        if (siblingNode->left->color == Black) {
+          // Color the *left* child of siblingNode Black, and siblingNode to Red
+          siblingNode->right->color = Black;
+          siblingNode->color = Red;
+
+          // rotateLeft(siblingNode) places the Red siblingNode as the *left*
+          // + child of the previous siblingNode->right
+          rotateLeft(siblingNode);
+          // Update the siblingNode of start node to reflect the new sibling
+          siblingNode = start->parent->left;
+          // After rotateRight(siblingNode), we have put ourselves into case 4
+        }
+
+        // siblingNode->left is colored Black, we are in case 4
+        siblingNode->color = start->parent->color;
+        start->parent->color = Black;
+        siblingNode->left->color = Black;
+
+        // rotateRight around the start->parent node, placing the siblingNode
+        // + In the previous position of the start->parent node
+        rotateRight(start->parent);
+
+        // Setting the start node to root node ensures the while() terminates
+        start = root;
+      }
+    }
+
+  }
+
+  // TODO: I could only get this working by using nil->parent
+  //   to store the parent of the last replacementNode within removeNode()
+  //   Is this expected, or have I missed something and hacked this?
+
+  // Update nil->parent to nullptr if it is any other value
+  // + When we transplant(), we temporarily store the parent of the relocatedNode
+  // + relocatedNode is seen passed to transplant() within remove()
+  if (nil->parent != nullptr) nil->parent = nullptr;
+}
+
 /** makeEmpty
- * @brief Recursively delete the given root BinaryNode and all of its children
+ * @brief Recursively delete the given root RedBlackNode and all of its children
  *
  * Runs in O( n ) time, since we need to visit each node in the tree once
  * + Where n is the total number of nodes in the tree
  *
- * @param tree The root BinaryNode to delete, along with all child nodes
+ * @param tree The root RedBlackNode to delete, along with all child nodes
  */
-void BinarySearchTree::makeEmpty(BinarySearchTree::BinaryNode * & tree)
+void RedBlackTree::makeEmpty(RedBlackTree::RedBlackNode * & tree)
 {
   // Base case: When all nodes have been deleted, tree is a nullptr
   // + Breaks from recursion
-  if (tree != nullptr) {
+  if (tree != nil) {
     makeEmpty(tree->left);
     makeEmpty(tree->right);
     delete tree;
-    tree = nullptr;
+    // Set each deleted node = nil, overwriting unused children with nullptr
+    tree = nil;
   }
+
 }
 
 /** isEmpty
- * @brief Determine whether or not the calling BST object is empty
+ * @brief Determine whether or not the calling RBT object is empty
  *
  * Runs in constant time, O( 1 )
  *
  * @return true If this->root node points to an empty tree (nullptr)
- * @return false If this->root node points to a constructed BinaryNode
+ * @return false If this->root node points to a constructed RedBlackNode
  */
-bool BinarySearchTree::isEmpty() const
+bool RedBlackTree::isEmpty() const
 {
-  return root == nullptr;
+  return root == nil;
 }
 
 /** insert
- * @brief Insert a value into the tree starting at a given BinaryNode
+ * @brief Insert a value into the tree starting at a given RedBlackNode
  * + Uses recursion
  *
- * Runs in O( height ) time, since in the worst case we insert the node at the
+ * Runs in O( lg(n) ) time, since the height of RBTs are <= lg(n)
- * + bottom of the BST.
+ * + Where n is the number of nodes in the RBT
+ * + The *sequential* call to insertFixup is also O( lg(n) )..
+ * ++ Appears to be O( 2lg(n) ), but we drop the constant 2; No extra overhead
  *
  * @param newValue The value to be inserted
- * @param start The BinaryNode to begin insertion
+ * @param start The RedBlackNode to begin insertion
- * @param prevNode The last checked BinaryNode
+ * @param prevNode The last checked RedBlackNode
  * + prevNode is used to initialize new node's parent
  */
-void BinarySearchTree::insert(const int &newValue,
+void RedBlackTree::insert(const int &newValue,
-                              BinaryNode *&start, BinaryNode *prevNode)
+                          RedBlackNode *&start, RedBlackNode *prevNode)
 {
   // Base case: We found a valid position which is empty for the newValue
-  if (start == nullptr) {
+  if (start == nil) {
     // Build a new node, place it at the current position
-    // + Breaks out of recursion
+    // + Breaks out of recursion after this code block finishes
-    start = new BinaryNode(newValue, nullptr, nullptr, prevNode);
+
+    // TODO: Valgrind thinks there is a memory leak here
+    //   It seems to me that calling `delete start` would delete the static nil?
+    start = new RedBlackNode(newValue, Red, nil, nil, prevNode);
+
+    // Enforce RBT properties on the entire subtree of start node
+    // + By default, start is the root node of the RBT unless specified
+    // + This is done with a call to the inlined insert(int) function
+    insertFixup(start);
   }
   else if (newValue < start->element) insert(newValue, start->left, start);
   else if (newValue > start->element) insert(newValue, start->right, start);
@@ -152,62 +484,103 @@ void BinarySearchTree::insert(const int &newValue,
 }
 
 /** remove
- * @brief Removes a value from the BST of the given BinaryNode
+ * @brief Removes a value from the RBT of the given RedBlackNode
  *
- * Runs in O( height ) time, where findMin() is the limiting function
+ * Runs in O( lg(n) ) time, since the height of RBTs are <= lg(n)
- * + remove() and transplant() otherwise run in constant time, without findMin()
+ * + Where n is the number of nodes in the RBT
+ * + The *sequential* call to removeFixup is also O( lg(n) )..
+ * ++ Appears to be O( 2lg(n) ), but we drop the constant 2; No extra overhead
  *
- * @param removeNode The BinaryNode to remove from the BST
+ * @param removeNode The RedBlackNode to remove from the RBT
  */
-void BinarySearchTree::remove(BinaryNode *removeNode)
+void RedBlackTree::remove(RedBlackNode *removeNode)
 {
-  if (removeNode->left == nullptr) {
+  RedBlackNode *replacementNode = removeNode;
+  Color originalColor = removeNode->color;
+
+  // This node will be passed to removeFixup() later to enforce RBT properties
+  RedBlackNode *fixupNode;
+
+  if (removeNode->left == nil) {
     // removeNode has no left node; Replace removeNode with removeNode->right
     // + It doesn't matter if removeNode->right is nullptr or a valid node
     // + Since there is no left node, this is the only possible valid transplant
 
     // Transplant the right node and its subtree over this node
+    fixupNode = removeNode->right;
     transplant(removeNode, removeNode->right);
   }
-  else if (removeNode->right == nullptr) {
+  else if (removeNode->right == nil) {
     // removeNode has no right node; Replace removeNode with removeNode->right
     // + removeNode->left exists, in this case
+    fixupNode = removeNode->left;
     transplant(removeNode, removeNode->left);
   }
   else {
-    // removeNode has a right and left node, find the next value in-order
+    // The node to remove has both left and right child nodes
-    // + findMin(removeNode->right) returns the next largest value in the BST
+    // + We should find a replacement within the child with a larger subtree
-    BinaryNode *minNode = findMin(removeNode->right);
 
-    // If the next value in-order is not removeNode->right
+    // Compare the depth of subtrees for both children of removeNode...
-    if (minNode->parent != removeNode) {
+    // + Red child node indicates the equal or longer subtree
-      // replace minNode with the next largest attached value, minNode->right
+    // + Black child node indicates the depth of the child's subtree is smaller
-      transplant(minNode, minNode->right);
+    if (removeNode->left->color) {
-      // Set minNode->right to the node at removeNode->right
+      // + findMax(removeNode->right) returns the next smallest value in the RBT
-      // + Update the parent of removeNode->right accordingly in the next line
+      replacementNode = findMax(removeNode->left);
-      minNode->right = removeNode->right;
+      originalColor = replacementNode->color;
-      minNode->right->parent = minNode;
+
+      fixupNode = replacementNode->right;
+
+      if (replacementNode->parent == removeNode) {
+        fixupNode->parent = replacementNode;
+      }
+      else {
+        transplant(replacementNode, replacementNode->right);
+        replacementNode->right = removeNode->right;
+        replacementNode->right->parent = replacementNode;
+      }
+      transplant(removeNode, replacementNode);
+      replacementNode->left = removeNode->left;
+      replacementNode->left->parent = replacementNode;
+      replacementNode->color = removeNode->color;
+    }
+    else {
+      // If the removeNode->left child is black, use the right subtree
+      // + findMin(removeNode->right) returns the next largest value in the RBT
+      replacementNode = findMin(removeNode->right);
+      originalColor = replacementNode->color;
+      fixupNode = replacementNode->right;
+
+      if (replacementNode->parent == removeNode) {
+        fixupNode->parent = replacementNode;
+      }
+      else {
+        transplant(replacementNode, replacementNode->right);
+        replacementNode->right = removeNode->right;
+        replacementNode->right->parent = replacementNode;
+      }
+      transplant(removeNode, replacementNode);
+      replacementNode->left = removeNode->left;
+      replacementNode->left->parent = replacementNode;
+      replacementNode->color = removeNode->color;
+
+    }
   }
 
-    // Replace removeNode with the next node in-order
+  // If originalColor of the replacementNode is black, enforce RBT properties
-    // + Update the minNode's left and parent nodes in the following lines
+  if (!originalColor) deleteFixup(fixupNode);
-    transplant(removeNode, minNode);
-    minNode->left = removeNode->left;
-    minNode->left->parent = minNode;
-  }
 }
 
 /** printInOrder
  * @brief Uses recursion to output left subtree, root node, then right subtrees
  *
  * Runs in O( n ) time, since we need to visit each node in the tree once
- * + Where n is the total number of nodes within the BST
+ * + Where n is the total number of nodes within the RBT
  *
- * @param start The root BinaryNode to begin the 'In Order' output
+ * @param start The root RedBlackNode to begin the 'In Order' output
  */
-void BinarySearchTree::printInOrder(BinaryNode *start) const
+void RedBlackTree::printInOrder(RedBlackNode *start) const
 {
-  if(start != nullptr) {
+  if(start != nil) {
     printInOrder(start->left);
     std::cout << start->element << " ";
     printInOrder(start->right);
@@ -218,13 +591,13 @@ void BinarySearchTree::printInOrder(BinaryNode *start) const
  * @brief Uses recursion to output left subtree, right subtree, then the root
  *
  * Runs in O( n ) time, since we need to visit each node in the tree once
- * + Where n is the total number of nodes within the BST
+ * + Where n is the total number of nodes within the RBT
  *
- * @param start The root BinaryNode to begin the 'Post Order' output
+ * @param start The root RedBlackNode to begin the 'Post Order' output
  */
-void BinarySearchTree::printPostOrder(BinaryNode *start) const
+void RedBlackTree::printPostOrder(RedBlackNode *start) const
 {
-  if (start != nullptr) {
+  if (start != nil) {
     printPostOrder(start->left);
     printPostOrder(start->right);
     std::cout << start->element << " ";
@@ -235,13 +608,13 @@ void BinarySearchTree::printPostOrder(BinaryNode *start) const
  * @brief Uses recursion to output the root, then left subtree, right subtrees
  *
  * Runs in O( n ) time, since we need to visit each node in the tree once
- * + Where n is the total number of nodes within the BST
+ * + Where n is the total number of nodes within the RBT
  *
- * @param start The root BinaryNode to begin the 'Pre Order' output
+ * @param start The root RedBlackNode to begin the 'Pre Order' output
  */
-void BinarySearchTree::printPreOrder(BinaryNode *start) const
+void RedBlackTree::printPreOrder(RedBlackNode *start) const
 {
-  if (start != nullptr) {
+  if (start != nil) {
     std::cout << start->element << " ";
     printPreOrder(start->left);
     printPreOrder(start->right);
@@ -252,64 +625,64 @@ void BinarySearchTree::printPreOrder(BinaryNode *start) const
  * @brief Search for a given value within a tree or subtree using recursion
  *
  * Runs in O( height ) time
- * + In the worst case, we are searching for a node at the bottom of the BST
+ * + In the worst case, we are searching for a node at the bottom of the RBT
  *
  * @param value The value to search for
  * @param start The node to start the search from; Can be a subtree
- * @return A pointer to the BinaryNode containing the value within the BST
+ * @return A pointer to the RedBlackNode containing the value within the RBT
  * + Returns nullptr if the node was not found
  */
-BinarySearchTree::BinaryNode *BinarySearchTree::search(
+RedBlackTree::RedBlackNode *RedBlackTree::search(
-    const int &value, BinaryNode *start) const
+    const int &value, RedBlackNode *start) const
 {
-  // Base case: If BST is empty, or holds the value we are searching for
+  // Base case: If RBT is empty, or holds the value we are searching for
   // + Breaks out of recursion
-  if (start == nullptr || start->element == value) return start;
+  if (start == nil || start->element == value) return start;
   else if (start->element < value) return search(value, start->right);
   else if (start->element > value) return search(value, start->left);
 }
 
 /** findMin
- * @brief Find the minimum value within the BST of the given BinaryNode
+ * @brief Find the minimum value within the RBT of the given RedBlackNode
  *        + This example uses a while loop; findMax uses recursion
  *
  * Runs in O( height ) time
- * + In the worst case, we traverse to to the left-most bottom of the BST
+ * + In the worst case, we traverse to to the left-most bottom of the RBT
  *
- * @param start The root BinaryNode to begin checking values
+ * @param start The root RedBlackNode to begin checking values
- * @return A pointer to the BinaryNode which contains the smallest value
+ * @return A pointer to the RedBlackNode which contains the smallest value
- * + Returns nullptr if BST is empty
+ * + Returns nullptr if RBT is empty
  */
-BinarySearchTree::BinaryNode * BinarySearchTree::findMin(BinaryNode *start) const
+RedBlackTree::RedBlackNode * RedBlackTree::findMin(RedBlackNode *start) const
 {
   // If our tree is empty
-  if (start == nullptr) return nullptr;
+  if (start == nil) return nullptr;
 
-  while (start->left != nullptr) start = start->left;
+  while (start->left != nil) start = start->left;
 
   // If current node has no smaller children, it is min
   return start;
 }
 
 /** findMax
- * @brief Find the maximum value within the BST of the given BinaryNode
+ * @brief Find the maximum value within the RBT of the given RedBlackNode
  *        + This example uses recursion; findMin uses a while loop
  *        ++ Both functions can be implemented using a loop or recursion
  *
  * Runs in O( height ) time
- * + In the worst case, we traverse to to the right-most bottom of the BST
+ * + In the worst case, we traverse to to the right-most bottom of the RBT
  *
- * @param start The root BinaryNode to begin checking values
+ * @param start The root RedBlackNode to begin checking values
- * @return A pointer to the BinaryNode which contains the largest value
+ * @return A pointer to the RedBlackNode which contains the largest value
- * + returns nullptr if BST is empty
+ * + returns nullptr if RBT is empty
  */
-BinarySearchTree::BinaryNode * BinarySearchTree::findMax(BinaryNode *start) const
+RedBlackTree::RedBlackNode * RedBlackTree::findMax(RedBlackNode *start) const
 {
   // If our tree is empty
-  if (start == nullptr) return nullptr;
+  if (start == nil) return nullptr;
 
   // Base case: If current node has no larger children, it is max; Break recursion
-  if (start->right == nullptr) return start;
+  if (start->right == nil) return start;
 
   // Move down the right side of our tree and check again
   return findMax(start->right);
@@ -319,19 +692,19 @@ BinarySearchTree::BinaryNode * BinarySearchTree::findMax(BinaryNode *start) cons
  * @brief Finds the previous value in-order from the value at a given startNode
  *
  *  Runs in O( height ) time
- *  + In the worst case we traverse to the bottom of the BST
+ *  + In the worst case we traverse to the bottom of the RBT
  *
  * @param startNode The node containing the value to find predecessor of
  * @return The node which is the predecessor of startNode
  */
-BinarySearchTree::BinaryNode * BinarySearchTree::predecessor(BinaryNode *startNode) const
+RedBlackTree::RedBlackNode * RedBlackTree::predecessor(RedBlackNode *startNode) const
 {
-  if (startNode->left != nullptr) return findMax(startNode->left);
+  if (startNode->left != nil) return findMax(startNode->left);
 
   // If startNode has a parent, walk up the tree until we reach the top
   // + If startNode has no parent, we set it to nullptr and return
-  BinaryNode *temp = startNode->parent;
+  RedBlackNode *temp = startNode->parent;
-  while (temp != nullptr && temp->left == startNode) {
+  while (temp != nil && temp->left == startNode) {
     startNode = temp;
     temp = temp->parent;
   }
@@ -342,20 +715,20 @@ BinarySearchTree::BinaryNode * BinarySearchTree::predecessor(BinaryNode *startNo
  * @brief Finds the next value in-order from the value at a given startNode
  *
  *  Runs in O( height ) time
- *  + In the worst case we traverse to the bottom of the BST
+ *  + In the worst case we traverse to the bottom of the RBT
  *
  * @param startNode The node containing the value to find successor of
  * @return The node which is the successor of startNode
  */
-BinarySearchTree::BinaryNode * BinarySearchTree::successor(BinaryNode *startNode) const
+RedBlackTree::RedBlackNode * RedBlackTree::successor(RedBlackNode *startNode) const
 {
   // If there is a right subtree, next value in-order is findMin(rightSubtree)
-  if (startNode->right != nullptr) return findMin(startNode->right);
+  if (startNode->right != nil) return findMin(startNode->right);
 
   // If startNode has a parent, walk up the tree until we reach the top
   // + If startNode has no parent, we set it to nullptr and return
-  BinaryNode *temp = startNode->parent;
+  RedBlackNode *temp = startNode->parent;
-  while (temp != nullptr && temp->right == startNode) {
+  while (temp != nil && temp->right == startNode) {
     startNode = temp;
     temp = temp->parent;
   }
@@ -368,20 +741,22 @@ BinarySearchTree::BinaryNode * BinarySearchTree::successor(BinaryNode *startNode
 *******************************************************************************/
 
 /** clone
- * @brief Clone a BST node and all its children using recursion
+ * @brief Clone a RBT node and all its children using recursion
  *
  * Runs in O( n ) time, since each node must be copied individually
  *
  * @param start The node to begin cloning from
- * @return A pointer to the BinaryNode which is root node of the copied tree
+ * @return A pointer to the RedBlackNode which is root node of the copied tree
  */
-BinarySearchTree::BinaryNode * BinarySearchTree::clone(BinaryNode *start)
+RedBlackTree::RedBlackNode * RedBlackTree::clone(RedBlackNode *start)
 {
+  if (root == nil) root = new RedBlackNode();
+
   // Base case: There is nothing to copy, break from recursion
-  if (start == nullptr) return nullptr;
+  if (start == nil) return nil;
 
   // Construct all child nodes through recursion, return root node
-  return new BinaryNode(start);
+  return new RedBlackNode(start);
 }
 
 /** transplant
@@ -394,12 +769,12 @@ BinarySearchTree::BinaryNode * BinarySearchTree::clone(BinaryNode *start)
  * @param oldNode The node to overwrite with newNode
  * @param newNode The new node to take the place of oldNode
  */
-void BinarySearchTree::transplant(BinaryNode *oldNode, BinaryNode *newNode)
+void RedBlackTree::transplant(RedBlackNode *oldNode, RedBlackNode *newNode)
 {
   // case 1: If oldNode is the root node at this->root
   // + 2: if the oldNode is the left child of it's parent
   // + 3: case if the oldNode  is the right child of  it's parent
-  if (oldNode->parent == nullptr) root = newNode;
+  if (oldNode->parent == nil) root = newNode;
   else if (oldNode == oldNode->parent->left) {
     // Update the parent of oldNode to reflect the transplant
     oldNode->parent->left = newNode;
@@ -410,5 +785,5 @@ void BinarySearchTree::transplant(BinaryNode *oldNode, BinaryNode *newNode)
   }
 
   // If we did not replace oldNode with a nullptr, update newNode's parent
-  if (newNode != nullptr) newNode->parent = oldNode->parent;
+  newNode->parent = oldNode->parent;
 }

cpp/algorithms/trees/redblack/redblack.h

@@ -13,71 +13,88 @@
 
 #include <iostream>
 
-// TODO: Add balance() method to balance overweight branches
+enum Color {Black, Red};
-class BinarySearchTree {
+
+class RedBlackTree {
 
 public:
-  // BinaryNode Structure
+  // RedBlackNode Structure
-  struct BinaryNode{
+  struct RedBlackNode{
     int element;
-    BinaryNode *left, *right, *parent;
+    Color color = Black;
+    RedBlackNode *left{}, *right{}, *parent{};
 
+    RedBlackNode() : element(INT32_MIN), color(Black) {}
     // Ctor for specific element, lhs, rhs
-    BinaryNode(const int &el, BinaryNode *lt, BinaryNode *rt, BinaryNode *p)
+    RedBlackNode(const int &el, Color c,
-        :element(el), left(lt), right(rt), parent(p) {};
+                 RedBlackNode *lt, RedBlackNode *rt, RedBlackNode *p)
-    // Ctor for a node and any downstream nodes
+        :element(el), color(c), left(lt), right(rt), parent(p) {};
-    explicit BinaryNode(BinaryNode * toCopy);
+    // Ctor for copying a node and any downstream nodes
+    explicit RedBlackNode(RedBlackNode * toCopy);
   };
+  static RedBlackNode *nil;
 
-  BinarySearchTree() : root(nullptr) {};
+  RedBlackTree() : root(nil) {};
-  BinarySearchTree(const BinarySearchTree &rhs) : root(rhs.clone(rhs.root)) {};
+  RedBlackTree(const RedBlackTree &rhs);;
-  BinarySearchTree& operator=(const BinarySearchTree& rhs);
+  RedBlackTree& operator=(const RedBlackTree& rhs);
-  ~BinarySearchTree() { makeEmpty(root);};
+  ~RedBlackTree() { makeEmpty(root);};
-  inline BinaryNode * getRoot() const { return root;}
+  // Inlined functions provide less verbose interface for using the RBT
+  inline RedBlackNode * getRoot() const { return root;}
+
+  void rotateLeft(RedBlackNode *pivotNode);
+  void rotateRight(RedBlackNode *pivotNode);
+  void insertFixup(RedBlackNode * start);
+  void deleteFixup(RedBlackNode * start);
 
   // Check if value is within the tree or subtree
   inline bool contains(const int &value) const { return contains(value, root);}
-  bool contains(const int &value, BinaryNode *start) const;
+  bool contains(const int &value, RedBlackNode *start) const;
 
   // Empties a given tree or subtree
   inline void makeEmpty() { makeEmpty(root);}
-  void makeEmpty(BinaryNode *&tree);
+  void makeEmpty(RedBlackNode *&tree);
-  // Checks if this BST is empty
+  // Checks if this RBT is empty
   bool isEmpty() const;
 
   // Insert and remove values from a tree or subtree
-  inline void insert(const int &x) { insert(x, root, nullptr);}
+  inline void insert(const int &x) { insert(x, root, nil);}
-  void insert(const int &newValue, BinaryNode *&start, BinaryNode *prevNode);
+  void insert(const int &newValue, RedBlackNode *&start, RedBlackNode *prevNode);
   inline void remove(const int &x) { remove(search(x, root));}
-  void remove(BinaryNode *removeNode);
+  void remove(RedBlackNode *removeNode);
 
   // Traversal functions
   inline void printInOrder() const { printInOrder(root);}
   inline void printPostOrder() const { printPostOrder(root);}
   inline void printPreOrder() const { printPreOrder(root);}
   // Overloaded to specify traversal of a subtree
-  void printInOrder(BinaryNode *start) const;
+  void printInOrder(RedBlackNode *start) const;
-  void printPostOrder(BinaryNode *start) const;
+  void printPostOrder(RedBlackNode *start) const;
-  void printPreOrder(BinaryNode *start) const;
+  void printPreOrder(RedBlackNode *start) const;
 
   // Find a BinaryNode containing value starting at a given tree / subtree node
-  inline BinaryNode * search(const int &value) const { return search(value, root);}
+  inline RedBlackNode * search(const int &value) const
-  BinaryNode * search(const int &value, BinaryNode *start) const;
+      { return search(value, root);}
-  inline BinaryNode * findMin() const { return findMin(root);}
+  RedBlackNode * search(const int &value, RedBlackNode *start) const;
-  inline BinaryNode * findMax() const { return findMax(root);}
-  // Find nodes with min / max values starting at a given tree / subtree node
-  BinaryNode * findMin(BinaryNode *start) const;
-  BinaryNode * findMax(BinaryNode *start) const;
 
-  BinaryNode * predecessor(BinaryNode *startNode) const;
+  inline RedBlackNode * findMin() const { return findMin(root);}
-  BinaryNode * successor(BinaryNode *startNode) const;
+  inline RedBlackNode * findMax() const { return findMax(root);}
+  // Find nodes with min / max values starting at a given tree / subtree node
+  RedBlackNode * findMin(RedBlackNode *start) const;
+  RedBlackNode * findMax(RedBlackNode *start) const;
+
+  inline RedBlackNode * predecessor(const int &value) const
+      { return predecessor(search(value));}
+  RedBlackNode * predecessor(RedBlackNode *startNode) const;
+  inline RedBlackNode * successor(const int &value) const
+      { return successor(search(value));}
+  RedBlackNode * successor(RedBlackNode *startNode) const;
 
 private:
-  // BST Private Member Functions
+  RedBlackNode * clone(RedBlackNode *start);
-  static BinaryNode * clone(BinaryNode *start);
+  void transplant(RedBlackNode *oldNode, RedBlackNode *newNode);
-  void transplant(BinaryNode *oldNode, BinaryNode *newNode);
 
-  BinaryNode *root;
+  // The root node for the RBT
+  RedBlackNode *root;
 };
 
 #endif // REDBLACK_H