En ciencias de la computación, un árbol binario de búsqueda es un árbol que tiene las siguientes propiedades:

• Cada nodo tiene un valor
• Se define un orden total sobre esos valores
• El subárbol izquierdo de un nodo contiene valores menores o iguales que el valor de dicho nodo.
• El subárbol derecho de un nodo contiene valores mayores o iguales que el valor de dicho nodo.

La ventaja más notable de los árboles binarios de búsqueda es que los algoritmos de ordenación y búsqueda relacionados como transversal inorden pueden ser muy eficientes.

Los árboles binarios de búsqueda son una estructura de datos fundamental usada para construir más estructuras de datos abstractas como conjuntos y arrays asociativos.
// BinarySearchTree class
//
// CONSTRUCTION: with no initializer
//
// ******************PUBLIC OPERATIONS*********************
// void insert( x ) --> Insert x
// void remove( x ) --> Remove x
// void removeMin( ) --> Remove minimum item
// 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
// ******************ERRORS********************************
// Exceptions are thrown by insert, remove, and removeMin if warranted

/**
* Implements an unbalanced binary search tree.
* Note that all "matching" is based on the compareTo method.
* @author Mark Allen Weiss
*/

public class BinarySearchTree {
/**
* Construct the tree.
*/
public BinarySearchTree( ) {
root = null;
}

/**
* Insert into the tree.
* @param x the item to insert.
* @throws DuplicateItemException if x is already present.
*/
public void insert( Comparable x ) {
root = insert( x, root );
}

/**
* Remove from the tree..
* @param x the item to remove.
* @throws ItemNotFoundException if x is not found.
*/
public void remove( Comparable x ) {
root = remove( x, root );
}

/**
* Remove minimum item from the tree.
* @throws ItemNotFoundException if tree is empty.
*/
public void removeMin( ) {
root = removeMin( root );
}

/**
* Find the smallest item in the tree.
* @return smallest item or null if empty.
*/
public Comparable findMin( ) {
return elementAt( findMin( root ) );
}

/**
* Find the largest item in the tree.
* @return the largest item or null if empty.
*/

public Comparable findMax( ) {
return elementAt( findMax( root ) );
}

/**
* 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 ) {
return elementAt( find( x, root ) );
}

/**
* Make the tree logically empty.
*/
public void makeEmpty( ) {
root = null;
}

/**
* Test if the tree is logically empty.
* @return true if empty, false otherwise.
*/
public boolean isEmpty( ) {
return root == null;
}

/**
* Internal method to get element field.
* @param t the node.
* @return the element field or null if t is null.
*/
private Comparable elementAt( BinaryNode t ) {
return t == null ? null : t.element;
}

/**
* Internal method to insert into a subtree.
* @param x the item to insert.
* @param t the node that roots the tree.
* @return the new root.
* @throws DuplicateItemException if x is already present.
*/
protected BinaryNode insert( Comparable x, BinaryNode t ) {
if( t == null )
t = new BinaryNode( x );
else if( x.compareTo( t.element ) < 0 )
t.left = insert( x, t.left );
else if( x.compareTo( t.element ) > 0 )
t.right = insert( x, t.right );
else
throw new DuplicateItemException( x.toString( ) ); // Duplicate
return t;
}

/**
* Internal method to remove from a subtree.
* @param x the item to remove.
* @param t the node that roots the tree.
* @return the new root.
* @throws ItemNotFoundException if x is not found.
*/
protected BinaryNode remove( Comparable x, BinaryNode t ) {
if( t == null )
throw new ItemNotFoundException( x.toString( ) );
if( x.compareTo( t.element ) < 0 )
t.left = remove( x, t.left );
else if( x.compareTo( t.element ) > 0 )
t.right = remove( x, t.right );
else if( t.left != null && t.right != null ) // Two children
{
t.element = findMin( t.right ).element;
t.right = removeMin( t.right );
} else
t = ( t.left != null ) ? t.left : t.right;
return t;
}

/**
* Internal method to remove minimum item from a subtree.
* @param t the node that roots the tree.
*@return the new root.
* @throws ItemNotFoundException if x is not found.
*/
protected BinaryNode removeMin( BinaryNode t ) {
if( t == null )
throw new ItemNotFoundException( );
else if( t.left != null ) {
t.left = removeMin( t.left );
return t;
} else
return t.right;
}

/**
* Internal method to find the smallest item in a subtree.
* @param t the node that roots the tree.
* @return node containing the smallest item.
*/
protected BinaryNode findMin( BinaryNode t ) {
if( t != null )
while( t.left != null )
t = t.left;

return t;
}

/**
* Internal method to find the largest item in a subtree.
* @param t the node that roots the tree.
* @return node containing the largest item.
*/
private BinaryNode findMax( BinaryNode t ) {
if( t != null )
while( t.right != null )
t = t.right;

return t;
}

/**
* Internal method to find an item in a subtree.
* @param x is item to search for.
* @param t the node that roots the tree.
* @return node containing the matched item.
*/
private BinaryNode find( Comparable x, BinaryNode t ) {
while( t != null ) {
if( x.compareTo( t.element ) < 0 )
t = t.left;
else if( x.compareTo( t.element ) > 0 )
t = t.right;
else
return t; // Match
}

return null; // Not found
}

/** The tree root. */
protected BinaryNode root;

// Test program
public static void main( String [ ] args ) {
BinarySearchTree t = new BinarySearchTree( );
final int NUMS = 4000;
final int GAP = 37;

System.out.println( "Checking... (no more output means success)" );

for( int i = GAP; i != 0; i = ( i + GAP ) % NUMS )
t.insert( new Integer( i ) );

for( int i = 1; i < NUMS; i+= 2 )
t.remove( new Integer( i ) );

if( ((Integer)(t.findMin( ))).intValue( ) != 2 ||
((Integer)(t.findMax( ))).intValue( ) != NUMS - 2 )
System.out.println( "FindMin or FindMax error!" );

for( int i = 2; i < NUMS; i+=2 )
if( ((Integer)(t.find( new Integer( i ) ))).intValue( ) != i )
System.out.println( "Find error1!" );

for( int i = 1; i < NUMS; i+=2 ) {
if( t.find( new Integer( i ) ) != null )
System.out.println( "Find error2!" );
}
}
}

// Basic node stored in unbalanced binary search trees
// Note that this class is not accessible outside
// of this package.

class BinaryNode {
// Constructors
BinaryNode( Comparable theElement ) {
element = theElement;
left = right = null;
}

// Friendly data; accessible by other package routines
Comparable element; // The data in the node
BinaryNode left; // Left child
BinaryNode right; // Right child
}

/**
* 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 );
}
}

/**
* Exception class for failed finds/removes in search
* trees, hash tables, and list and tree iterators.
* @author Mark Allen Weiss
*/
public class ItemNotFoundException extends RuntimeException {
/**
* Construct this exception object.
*/
public ItemNotFoundException( ) {
super( );
}

/**
* Construct this exception object.
* @param message the error message.
*/
public ItemNotFoundException( String message ) {
super( message );
}
}

Fuente: Java-Tips.org, http://www.java-tips.org/java-se-tips/java.lang/binary-search-tree-implementation-in-java.htm