A red-black tree is a type of self-balancing binary search tree, a data structure used in computer science, typically used to implement associative arrays. The original structure was invented in 1972 by Rudolf Bayer who called them "symmetric binary B-trees", but acquired its modern name in a paper in 1978 by Leo J. Guibas and Robert Sedgewick. It is complex, but has good worst-case running time for its operations and is efficient in practice: it can search, insert, and delete in O(log n) time, where n is the number of elements in the tree.

A red-black tree is a special type of binary tree, which is a structure used in computer science to organize pieces of comparable data, such as numbers. Each piece of data is stored in a node. One of the nodes always functions as our starting place, and is not the child of any node; we call this the root node or root. It has up to two "children", other nodes to which it connects. Each of these children can have children of its own, and so on. The root node thus has a path connecting it to any other node in the tree.

If a node has no children, we call it a leaf node, since intuitively it is at the periphery of the tree. A subtree is the portion of the tree that can be reached from a certain node, considered as a tree itself. In red-black trees, the leaves are assumed to be null or empty.

As red-black trees are also binary search trees, they must satisfy the constraint that every node contains a value greater than or equal to all nodes in its left subtree, and less than or equal to all nodes in its right subtree. This makes it quick to search the tree for a given value.

The following code shows how to implement a red-black tree in Java:

 ``` // RedBlackTree class // // CONSTRUCTION: with a negative infinity sentinel // // ******************PUBLIC OPERATIONS********************* // void insert( x ) --> Insert x // void remove( x ) --> Remove x (unimplemented) // Comparable find( x ) --> Return item that matches x // Comparable findMin( ) --> Return smallest item // Comparable findMax( ) --> Return largest item // boolean isEmpty( ) --> Return true if empty; else false // void makeEmpty( ) --> Remove all items // void printTree( ) --> Print all items // ******************ERRORS******************************** // Exceptions are thrown by insert if warranted and remove. /** * Implements a red-black tree. * Note that all "matching" is based on the compareTo method. * @author Mark Allen Weiss */ public class RedBlackTree { /** * Construct the tree. */ public RedBlackTree( ) { header = new RedBlackNode( null ); header.left = header.right = nullNode; } /** * Compare item and t.element, using compareTo, with * caveat that if t is header, then item is always larger. * This routine is called if is possible that t is header. * If it is not possible for t to be header, use compareTo directly. */ private final int compare( Comparable item, RedBlackNode t ) { if( t == header ) return 1; else return item.compareTo( t.element ); } /** * Insert into the tree. * @param item the item to insert. * @throws DuplicateItemException if item is already present. */ public void insert( Comparable item ) { current = parent = grand = header; nullNode.element = item; while( compare( item, current ) != 0 ) { great = grand; grand = parent; parent = current; current = compare( item, current ) < 0 ? current.left : current.right; // Check if two red children; fix if so if( current.left.color == RED && current.right.color == RED ) handleReorient( item ); } // Insertion fails if already present if( current != nullNode ) throw new DuplicateItemException( item.toString( ) ); current = new RedBlackNode( item, nullNode, nullNode ); // Attach to parent if( compare( item, parent ) < 0 ) parent.left = current; else parent.right = current; handleReorient( item ); } /** * Remove from the tree. * @param x the item to remove. * @throws UnsupportedOperationException if called. */ public void remove( Comparable x ) { throw new UnsupportedOperationException( ); } /** * Find the smallest item the tree. * @return the smallest item or null if empty. */ public Comparable findMin( ) { if( isEmpty( ) ) return null; RedBlackNode itr = header.right; while( itr.left != nullNode ) itr = itr.left; return itr.element; } /** * Find the largest item in the tree. * @return the largest item or null if empty. */ public Comparable findMax( ) { if( isEmpty( ) ) return null; RedBlackNode itr = header.right; while( itr.right != nullNode ) itr = itr.right; return itr.element; } /** * Find an item in the tree. * @param x the item to search for. * @return the matching item or null if not found. */ public Comparable find( Comparable x ) { nullNode.element = x; current = header.right; for( ; ; ) { if( x.compareTo( current.element ) < 0 ) current = current.left; else if( x.compareTo( current.element ) > 0 ) current = current.right; else if( current != nullNode ) return current.element; else return null; } } /** * Make the tree logically empty. */ public void makeEmpty( ) { header.right = nullNode; } /** * Print all items. */ public void printTree( ) { printTree( header.right ); } /** * Internal method to print a subtree in sorted order. * @param t the node that roots the tree. */ private void printTree( RedBlackNode t ) { if( t != nullNode ) { printTree( t.left ); System.out.println( t.element ); printTree( t.right ); } } /** * Test if the tree is logically empty. * @return true if empty, false otherwise. */ public boolean isEmpty( ) { return header.right == nullNode; } /** * Internal routine that is called during an insertion * if a node has two red children. Performs flip and rotations. * @param item the item being inserted. */ private void handleReorient( Comparable item ) { // Do the color flip current.color = RED; current.left.color = BLACK; current.right.color = BLACK; if( parent.color == RED ) // Have to rotate { grand.color = RED; if( ( compare( item, grand ) < 0 ) != ( compare( item, parent ) < 0 ) ) parent = rotate( item, grand ); // Start dbl rotate current = rotate( item, great ); current.color = BLACK; } header.right.color = BLACK; // Make root black } /** * Internal routine that performs a single or double rotation. * Because the result is attached to the parent, there are four cases. * Called by handleReorient. * @param item the item in handleReorient. * @param parent the parent of the root of the rotated subtree. * @return the root of the rotated subtree. */ private RedBlackNode rotate( Comparable item, RedBlackNode parent ) { if( compare( item, parent ) < 0 ) return parent.left = compare( item, parent.left ) < 0 ? rotateWithLeftChild( parent.left ) : // LL rotateWithRightChild( parent.left ) ; // LR else return parent.right = compare( item, parent.right ) < 0 ? rotateWithLeftChild( parent.right ) : // RL rotateWithRightChild( parent.right ); // RR } /** * Rotate binary tree node with left child. */ private static RedBlackNode rotateWithLeftChild( RedBlackNode k2 ) { RedBlackNode k1 = k2.left; k2.left = k1.right; k1.right = k2; return k1; } /** * Rotate binary tree node with right child. */ private static RedBlackNode rotateWithRightChild( RedBlackNode k1 ) { RedBlackNode k2 = k1.right; k1.right = k2.left; k2.left = k1; return k2; } private static class RedBlackNode { // Constructors RedBlackNode( Comparable theElement ) { this( theElement, null, null ); } RedBlackNode( Comparable theElement, RedBlackNode lt, RedBlackNode rt ) { element = theElement; left = lt; right = rt; color = RedBlackTree.BLACK; } Comparable element; // The data in the node RedBlackNode left; // Left child RedBlackNode right; // Right child int color; // Color } private RedBlackNode header; private static RedBlackNode nullNode; static // Static initializer for nullNode { nullNode = new RedBlackNode( null ); nullNode.left = nullNode.right = nullNode; } private static final int BLACK = 1; // Black must be 1 private static final int RED = 0; // Used in insert routine and its helpers private static RedBlackNode current; private static RedBlackNode parent; private static RedBlackNode grand; private static RedBlackNode great; // Test program public static void main( String [ ] args ) { RedBlackTree t = new RedBlackTree( ); final int NUMS = 400000; final int GAP = 35461; System.out.println( "Checking... (no more output means success)" ); for( int i = GAP; i != 0; i = ( i + GAP ) % NUMS ) t.insert( new Integer( i ) ); if( ((Integer)(t.findMin( ))).intValue( ) != 1 || ((Integer)(t.findMax( ))).intValue( ) != NUMS - 1 ) System.out.println( "FindMin or FindMax error!" ); for( int i = 1; i < NUMS; i++ ) if( ((Integer)(t.find( new Integer( i ) ))).intValue( ) != i ) System.out.println( "Find error1!" ); } } /** * Exception class for duplicate item errors * in search tree insertions. * @author Mark Allen Weiss */ public class DuplicateItemException extends RuntimeException { /** * Construct this exception object. */ public DuplicateItemException( ) { super( ); } /** * Construct this exception object. * @param message the error message. */ public DuplicateItemException( String message ) { super( message ); } } ```