代码出处

https://users.cs.fiu.edu/~weiss/

各种实现

• Java实现

Data Structures and Algorithm Analysis in Java (Third Edition)
CS-7 Text
https://users.cs.fiu.edu/~weiss/dsaajava3/code/
Fig02_09.java: Test program for binary search
BinarySearchTree.java: Binary search tree

测试运行

• 源码文件：BinarySearchTree.java 以及Fig02_09.java（放置到同一个文件夹下面）
• 编译运行：

> cd bst
> ls
BinarySearchTree.java
Fig02_09.java

> javac Fig02_09.java
> java Fig02_09
Found 0 at 0
Found 1 at -1
Found 2 at 1
Found 3 at -1
Found 4 at 2
Found 5 at -1
Found 6 at 3
Found 7 at -1
Found 8 at 4
Found 9 at -1
Found 10 at 5
Found 11 at -1
Found 12 at 6
Found 13 at -1
Found 14 at 7
Found 15 at -1

算法分析

删除操作 remove

3种情况

• Case 1 : 要删除的结点，本身是一个叶子结点，直接删除即可；

• Case 2：要删除的结点，只有一个孩子，让其父结点指向其孩子结点即可；
  

  删除 结点4，结点4只有一个孩子结点

• Case 3：要删除的结点，有两个孩子，先用其右子树的最小数据替换该结点，然后递归地删除那个最小数据的结点（这个最小数据结点一定是没有左孩子的）；
  

  删除 结点2，结点2 有两个孩子结点

remove 实现

    /**
     * Internal method to remove from a subtree.
     * @param x the item to remove.
     * @param t the node that roots the subtree.
     * @return the new root of the subtree.
     */
    private BinaryNode<AnyType> remove( AnyType x, BinaryNode<AnyType> t )
    {
        if( t == null )
            return t;   // Item not found; do nothing
            
        int compareResult = x.compareTo( t.element );
            
        if( compareResult < 0 )
            t.left = remove( x, t.left );
        else if( compareResult > 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 = remove( t.element, t.right );
        }
        else
            t = ( t.left != null ) ? t.left : t.right;
        return t;
    }

    /**
     * Internal method to find the smallest item in a subtree.
     * @param t the node that roots the subtree.
     * @return node containing the smallest item.
     */
    private BinaryNode<AnyType> findMin( BinaryNode<AnyType> t )
    {
        if( t == null )
            return null;
        else if( t.left == null )
            return t;
        return findMin( t.left );
    }

   

完整源码

测试代码 Fig02_09.java

public class Fig02_09
{
    public static final int NOT_FOUND = -1;

    /**
     * Performs the standard binary search.
     * @return index where item is found, or -1 if not found
     */
    public static <AnyType extends Comparable<? super AnyType>>
    int binarySearch( AnyType [ ] a, AnyType x )
    {
        int low = 0, high = a.length - 1;

        while( low <= high )
        {
            int mid = ( low + high ) / 2;

            if( a[ mid ].compareTo( x ) < 0 )
                low = mid + 1;
            else if( a[ mid ].compareTo( x ) > 0 )
                high = mid - 1;
            else
                return mid;   // Found
        }
        return NOT_FOUND;     // NOT_FOUND is defined as -1
    }

    // Test program
    public static void main( String [ ] args )
    {
        int SIZE = 8;
        Integer [ ] a = new Integer [ SIZE ];
        for( int i = 0; i < SIZE; i++ )
            a[ i ] = i * 2;

        for( int i = 0; i < SIZE * 2; i++ )
            System.out.println( "Found " + i + " at " + binarySearch( a, i ) );
    }
}

BST BinarySearchTree.java

// BinarySearchTree class
//
// CONSTRUCTION: with no initializer
//
// ******************PUBLIC OPERATIONS*********************
// void insert( x )       --> Insert x
// void remove( x )       --> Remove x
// boolean contains( x )  --> Return true if x is present
// 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 tree in sorted order
// ******************ERRORS********************************
// Throws UnderflowException as appropriate

/**
 * Implements an unbalanced binary search tree.
 * Note that all "matching" is based on the compareTo method.
 * @author Mark Allen Weiss
 */
public class BinarySearchTree<AnyType extends Comparable<? super AnyType>>
{
    /**
     * Construct the tree.
     */
    public BinarySearchTree( )
    {
        root = null;
    }

    /**
     * Insert into the tree; duplicates are ignored.
     * @param x the item to insert.
     */
    public void insert( AnyType x )
    {
        root = insert( x, root );
    }

    /**
     * Remove from the tree. Nothing is done if x is not found.
     * @param x the item to remove.
     */
    public void remove( AnyType x )
    {
        root = remove( x, root );
    }

    /**
     * Find the smallest item in the tree.
     * @return smallest item or null if empty.
     */
    public AnyType findMin( )
    {
        if( isEmpty( ) )
            throw new UnderflowException( );
        return findMin( root ).element;
    }

    /**
     * Find the largest item in the tree.
     * @return the largest item of null if empty.
     */
    public AnyType findMax( )
    {
        if( isEmpty( ) )
            throw new UnderflowException( );
        return findMax( root ).element;
    }

    /**
     * Find an item in the tree.
     * @param x the item to search for.
     * @return true if not found.
     */
    public boolean contains( AnyType x )
    {
        return contains( 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;
    }

    /**
     * Print the tree contents in sorted order.
     */
    public void printTree( )
    {
        if( isEmpty( ) )
            System.out.println( "Empty tree" );
        else
            printTree( root );
    }

    /**
     * Internal method to insert into a subtree.
     * @param x the item to insert.
     * @param t the node that roots the subtree.
     * @return the new root of the subtree.
     */
    private BinaryNode<AnyType> insert( AnyType x, BinaryNode<AnyType> t )
    {
        if( t == null )
            return new BinaryNode<>( x, null, null );
        
        int compareResult = x.compareTo( t.element );
            
        if( compareResult < 0 )
            t.left = insert( x, t.left );
        else if( compareResult > 0 )
            t.right = insert( x, t.right );
        else
            ;  // Duplicate; do nothing
        return t;
    }

    /**
     * Internal method to remove from a subtree.
     * @param x the item to remove.
     * @param t the node that roots the subtree.
     * @return the new root of the subtree.
     */
    private BinaryNode<AnyType> remove( AnyType x, BinaryNode<AnyType> t )
    {
        if( t == null )
            return t;   // Item not found; do nothing
            
        int compareResult = x.compareTo( t.element );
            
        if( compareResult < 0 )
            t.left = remove( x, t.left );
        else if( compareResult > 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 = remove( t.element, t.right );
        }
        else
            t = ( t.left != null ) ? t.left : t.right;
        return t;
    }

    /**
     * Internal method to find the smallest item in a subtree.
     * @param t the node that roots the subtree.
     * @return node containing the smallest item.
     */
    private BinaryNode<AnyType> findMin( BinaryNode<AnyType> t )
    {
        if( t == null )
            return null;
        else if( t.left == null )
            return t;
        return findMin( t.left );
    }

    /**
     * Internal method to find the largest item in a subtree.
     * @param t the node that roots the subtree.
     * @return node containing the largest item.
     */
    private BinaryNode<AnyType> findMax( BinaryNode<AnyType> 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 subtree.
     * @return node containing the matched item.
     */
    private boolean contains( AnyType x, BinaryNode<AnyType> t )
    {
        if( t == null )
            return false;
            
        int compareResult = x.compareTo( t.element );
            
        if( compareResult < 0 )
            return contains( x, t.left );
        else if( compareResult > 0 )
            return contains( x, t.right );
        else
            return true;    // Match
    }

    /**
     * Internal method to print a subtree in sorted order.
     * @param t the node that roots the subtree.
     */
    private void printTree( BinaryNode<AnyType> t )
    {
        if( t != null )
        {
            printTree( t.left );
            System.out.println( t.element );
            printTree( t.right );
        }
    }

    /**
     * Internal method to compute height of a subtree.
     * @param t the node that roots the subtree.
     */
    private int height( BinaryNode<AnyType> t )
    {
        if( t == null )
            return -1;
        else
            return 1 + Math.max( height( t.left ), height( t.right ) );    
    }
    
    // Basic node stored in unbalanced binary search trees
    private static class BinaryNode<AnyType>
    {
            // Constructors
        BinaryNode( AnyType theElement )
        {
            this( theElement, null, null );
        }

        BinaryNode( AnyType theElement, BinaryNode<AnyType> lt, BinaryNode<AnyType> rt )
        {
            element  = theElement;
            left     = lt;
            right    = rt;
        }

        AnyType element;            // The data in the node
        BinaryNode<AnyType> left;   // Left child
        BinaryNode<AnyType> right;  // Right child
    }


      /** The tree root. */
    private BinaryNode<AnyType> root;


        // Test program
    public static void main( String [ ] args )
    {
        BinarySearchTree<Integer> 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( i );

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

        if( NUMS < 40 )
            t.printTree( );
        if( t.findMin( ) != 2 || t.findMax( ) != NUMS - 2 )
            System.out.println( "FindMin or FindMax error!" );

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

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