算法设计和数据结构学习_5(BST&AVL&红黑树简单介绍)

　　前言：

　　节主要是给出BST，AVL和红黑树的C++代码，方便自己以后的查阅，其代码依旧是data structures and algorithm analysis in c++ (second edition)一书的作者所给，关于这3中二叉树在前面的博文算法设计和数据结构学习_4(《数据结构和问题求解》part4笔记)中已经有所介绍。这里不会去详细介绍它们的实现和规则，一是因为这方面的介绍性资料超非常多，另外这3种树的难点都在插入和删除部分，其规则本身并不多，但是要用文字和图形解释其实还蛮耗时的。所以，我们在看教程时，主要是要抓住这几种树的思想，然后对照对应的代码来看就ok了，能把代码看懂基本也就理解这些树的本质了。

　　BST& AVL树：

　　BST即二叉搜索树，它只需满足A节点左子树的值都小于A的值，右子树的值都大于A节点的值。其插入过程是依照它的属性值依次插入，删除过程分2种情况，如果是叶子节点，直接删除，如果是非叶子节点，则删除后将它的左子树中的最大节点填补，如果左子树为空，则用右子树中的最小节点填补。

　　AVL树的构造过程中有下面四种情况需要调整，有可能只需旋转一次，有可能需要旋转2次。

　　1. 单向右旋转（不平衡节点）平衡处理：

　　当在左子树上插入左节点，使平衡因子由1增加至2时。

　　2. 单向左旋转（不平衡节点）平衡处理：

　　当在右子树上插入右节点，使平衡因子由-1增加至-2时。

　　3. 双向旋转（先左旋转不平衡节点左孩子，然后右旋转不平衡节点）平衡处理：

　　当在左子树上插入右节点，使平衡因子有1增加到2时。

　　4. 双向旋转(先右旋转不平衡节点右孩子，然后左旋转不平衡节点)平衡处理：

　　当在右子树上插入左节点，使平衡因子由-1增加至-2时。

　　BST类实现的code如下(AVL类似)：

BinarySearchTree.h:

#ifndef BINARY_SEARCH_TREE_H_
#define BINARY_SEARCH_TREE_H_

#include "Wrapper.h"

template <class Comparable>
class BinarySearchTree;

template <class Comparable>
class BinarySearchTreeWithRank;

template <class Comparable>
class BinaryNode
{
    Comparable  element;
    BinaryNode *left;
    BinaryNode *right;
    int size;

    BinaryNode( const Comparable & theElement, BinaryNode *lt,
                BinaryNode *rt, int sz =  )
      : element( theElement ), left( lt ), right( rt ), size( sz ) { }

    friend class BinarySearchTree<Comparable>;
    friend class BinarySearchTreeWithRank<Comparable>;
};

// BinarySearchTree class
//
// CONSTRUCTION: with no parameters or another BinarySearchTree.
//
// ******************PUBLIC OPERATIONS*********************
// void insert( x )       --> Insert x
// void remove( x )       --> Remove x
// void removeMin( )      --> Remove smallest item
// Comparable find( x )   --> Return item that matches x
// Comparable findMin( )  --> Return smallest item
// Comparable findMax( )  --> Return largest item
// bool isEmpty( )        --> Return true if empty; else false
// void makeEmpty( )      --> Remove all items
// ******************ERRORS********************************
// Exceptions are thrown by insert, remove, and removeMin if warranted

template <class Comparable>
class BinarySearchTree
{
  public:
    BinarySearchTree( );
    BinarySearchTree( const BinarySearchTree & rhs );
    virtual ~BinarySearchTree( );

    Cref<Comparable> findMin( ) const;
    Cref<Comparable> findMax( ) const;
    Cref<Comparable> find( const Comparable & x ) const;
    bool isEmpty( ) const;

    void makeEmpty( );
    void insert( const Comparable & x );
    void remove( const Comparable & x );
    void removeMin( );

    const BinarySearchTree & operator=( const BinarySearchTree & rhs );

    typedef BinaryNode<Comparable> Node;

  protected:
    Node *root;

    Cref<Comparable> elementAt( Node *t ) const;
    virtual void insert( const Comparable & x, Node * & t ) const;
    virtual void remove( const Comparable & x, Node * & t ) const;
    virtual void removeMin( Node * & t ) const;
    Node * findMin( Node *t ) const;
    Node * findMax( Node *t ) const;
    Node * find( const Comparable & x, Node *t ) const;
    void makeEmpty( Node * & t ) const;
    Node * clone( Node *t ) const;
};

// BinarySearchTreeWithRank class.
//
// CONSTRUCTION: with no parameters or
//    another BinarySearchTreeWithRank.
//
// ******************PUBLIC OPERATIONS*********************
// Comparable findKth( k )--> Return kth smallest item
// All other operations are inherited (but C++ requires
// some extra stuff).

template <class Comparable>
class BinarySearchTreeWithRank : public BinarySearchTree<Comparable>
{
  public:
    Cref<Comparable> findKth( int k ) const;

    void insert( const Comparable & x )
      { BinarySearchTree<Comparable>::insert( x ); }
    void remove( const Comparable & x )
      { BinarySearchTree<Comparable>::remove( x ); }
    void removeMin( )
      { BinarySearchTree<Comparable>::removeMin( ); }

    typedef BinaryNode<Comparable> Node;

  private:
    void insert( const Comparable & x, Node * & t ) const;
    void remove( const Comparable & x, Node * & t ) const;
    void removeMin( Node * & t ) const;
    Node *findKth( int k, Node *t ) const;

    int treeSize( Node *t ) const
      { return t == NULL ?  : t->size; }
};

#include "BinarySearchTree.cpp"
#endif

BinarySearchTree.cpp:

#include "BinarySearchTree.h"
#include "Except.h"

// Construct the tree.
template <class Comparable>
BinarySearchTree<Comparable>::BinarySearchTree( ) : root( NULL )
{
}

// Copy constructor.
template <class Comparable>
BinarySearchTree<Comparable>::
BinarySearchTree( const BinarySearchTree<Comparable> & rhs ) : root( NULL )
{
    *this = rhs;
}

// Destructor for the tree.
template <class Comparable>
BinarySearchTree<Comparable>::~BinarySearchTree( )
{
    makeEmpty( );
}

// Insert x into the tree;
// Throws DuplicateItemException if x is already there.
template <class Comparable>
void BinarySearchTree<Comparable>::insert( const Comparable & x )
{
    insert( x, root );
}

// Remove x from the tree.
// Throws ItemNotFoundException if x is not in the tree.
template <class Comparable>
void BinarySearchTree<Comparable>::remove( const Comparable & x )
{
    remove( x, root );
}

// Remove minimum item from the tree.
// Throws UnderflowException if tree is empty.
template <class Comparable>
void BinarySearchTree<Comparable>::removeMin( )
{
    removeMin( root );
}

// Return the smallest item in the tree wrapped in a Cref object.
template <class Comparable>
Cref<Comparable> BinarySearchTree<Comparable>::findMin( ) const
{
    return elementAt( findMin( root ) );
}

// Return the largest item in the tree wrapped in a Cref object.
template <class Comparable>
Cref<Comparable> BinarySearchTree<Comparable>::findMax( ) const
{
    return elementAt( findMax( root ) );
}

// Find item x in the tree.
// Return the matching item wrapped in a Cref object.
template <class Comparable>
Cref<Comparable> BinarySearchTree<Comparable>::find( const Comparable & x ) const
{
    return elementAt( find( x, root ) );
}

// Make the tree logically empty.
template <class Comparable>
void BinarySearchTree<Comparable>::makeEmpty( )
{
    makeEmpty( root );
}

// Test if the tree is logically empty.
// Return true if empty, false otherwise.
template <class Comparable>
bool BinarySearchTree<Comparable>::isEmpty( ) const
{
    return root == NULL;
}

// Deep copy.
template <class Comparable>
const BinarySearchTree<Comparable> &
BinarySearchTree<Comparable>::
operator=( const BinarySearchTree<Comparable> & rhs )
{
    if( this != &rhs )
    {
        makeEmpty( );
        root = clone( rhs.root );
    }
    return *this;
}

// Internal method to wrap the element field in node t inside a Cref object.
template <class Comparable>
Cref<Comparable> BinarySearchTree<Comparable>::elementAt( Node *t ) const
{
    if( t == NULL )
        return Cref<Comparable>( );
    else
        return Cref<Comparable>( t->element );
}

// Internal method to insert into a subtree.
// x is the item to insert.
// t is the node that roots the tree.
// Set the new root.
// Throw DuplicateItemException if x is already in t.
template <class Comparable>
void BinarySearchTree<Comparable>::
insert( const Comparable & x, Node * & t ) const
{
    if( t == NULL )
        t = new Node( x, NULL, NULL );
    else if( x < t->element )
        insert( x, t->left );
    else if( t->element < x )
        insert( x, t->right );
    else
        throw DuplicateItemException( );
}

// Internal method to remove from a subtree.
// x is the item to remove.
// t is the node that roots the tree.
// Set the new root.
// Throw ItemNotFoundException is x is not in t.
template <class Comparable>
void BinarySearchTree<Comparable>::
remove( const Comparable & x, Node * & t ) const
{
    if( t == NULL )
        throw ItemNotFoundException( );
    if( x < t->element )
        remove( x, t->left );
    else if( t->element < x )
        remove( x, t->right );
    else if( t->left != NULL && t->right != NULL ) // Two children
    {
        t->element = findMin( t->right )->element;
        removeMin( t->right );                   // Remove minimum
    }
    else
    {
        BinaryNode<Comparable> *oldNode = t;
        t = ( t->left != NULL ) ? t->left : t->right;  // Reroot t
        delete oldNode;                         // delete old root
    }
}

// Internal method to remove minimum item from a subtree.
// t is the node that roots the tree.
// Set the new root.
// Throw UnderflowException if t is empty.
template <class Comparable>
void BinarySearchTree<Comparable>::removeMin( Node * & t ) const
{
    if( t == NULL )
        throw UnderflowException( );
    else if( t->left != NULL )
        removeMin( t->left );
    else
    {
        Node *tmp = t;
        t = t->right;
        delete tmp;
    }
}

// Internal method to find the smallest item in a subtree t.
// Return node containing the smallest item.
template <class Comparable>
BinaryNode<Comparable> * BinarySearchTree<Comparable>::findMin( Node *t ) const
{
    if( t != NULL )
        while( t->left != NULL )
            t = t->left;

    return t;
}

// Internal method to find the largest item in a subtree t.
// Return node containing the largest item.
template <class Comparable>
BinaryNode<Comparable> * BinarySearchTree<Comparable>::findMax( Node *t ) const
{
    if( t != NULL )
        while( t->right != NULL )
            t = t->right;

    return t;
}

// Internal method to find an item in a subtree.
// x is item to search for.
// t is the node that roots the tree.
// Return node containing the matched item.
template <class Comparable>
BinaryNode<Comparable> * BinarySearchTree<Comparable>::
find( const Comparable & x, Node *t ) const
{
    while( t != NULL )
        if( x < t->element )
            t = t->left;
        else if( t->element < x )
            t = t->right;
        else
            return t;    // Match

    return NULL;         // Not found
}

// Internal method to make subtree empty.
template <class Comparable>
void BinarySearchTree<Comparable>::makeEmpty( Node * & t ) const
{
    if( t != NULL )
    {
        makeEmpty( t->left );
        makeEmpty( t->right );
        delete t;
    }
    t = NULL;
}

// Internal method to clone subtree.
template <class Comparable>
BinaryNode<Comparable> * BinarySearchTree<Comparable>::clone( Node * t ) const
{
    if( t == NULL )
        return NULL;
    else
        return new Node( t->element, clone( t->left ), clone( t->right ), t->size );
}

// Returns the kth smallest item in the tree.
// Throws ItemNotFoundException if k is out of range.
template <class Comparable>
Cref<Comparable> BinarySearchTreeWithRank<Comparable>::findKth( int k ) const
{
    return elementAt( findKth( k, root ) );
}

// Internal method to insert into a subtree.
// x is the item to insert.
// t is the node that roots the tree.
// Set the new root.
// Throw DuplicateItemException if x is already in t.
template <class Comparable>
void BinarySearchTreeWithRank<Comparable>::
insert( const Comparable & x, Node * & t ) const
{
    if( t == NULL )
        t = new Node( x, NULL, NULL,  );
    else if( x < t->element )
        insert( x, t->left );
    else if( t->element < x )
        insert( x, t->right );
    else
        throw DuplicateItemException( );

    t->size++;
}

// Internal method to remove from a subtree.
// x is the item to remove.
// t is the node that roots the tree.
// Set the new root.
// Throw ItemNotFoundException is x is not in t.
template <class Comparable>
void BinarySearchTreeWithRank<Comparable>::
remove( const Comparable & x, Node * & t ) const
{
    if( t == NULL )
        throw ItemNotFoundException( );
    if( x < t->element )
        remove( x, t->left );
    else if( t->element < x )
        remove( x, t->right );
    else if( t->left != NULL && t->right != NULL ) // Two children
    {
        t->element = findMin( t->right )->element;
        removeMin( t->right );                   // Remove minimum
    }
    else
    {
        BinaryNode<Comparable> *oldNode = t;
        t = ( t->left != NULL ) ? t->left : t->right;  // Reroot t
        delete oldNode;                         // delete old root
        return;
    }

    t->size--;
}

// Internal method to remove minimum item from a subtree.
// t is the node that roots the tree.
// Set the new root.
// Throw UnderflowException if t is empty.
template <class Comparable>
void BinarySearchTreeWithRank<Comparable>::removeMin( Node * & t ) const
{
    if( t == NULL )
        throw UnderflowException( );
    else if( t->left != NULL )
        removeMin( t->left );
    else
    {
        Node *tmp = t;
        t = t->right;
        delete tmp;
        return;
    }

    t->size--;
}

// Internal method to find kth item in a subtree.
// k is the desired rank.
// t is the node that roots the tree.
template <class Comparable>
BinaryNode<Comparable> *
BinarySearchTreeWithRank<Comparable>::findKth( int k, Node * t ) const
{
    if( t == NULL )
        return NULL;

    int leftSize = treeSize( t->left );

    if( k <= leftSize )
        return findKth( k, t->left );
    else if( k == leftSize +  )
        return t;
    else
        return findKth( k - leftSize - , t->right );
}

　　红黑树：

　　3个连续的节点构成的树不可能是Red-Black树。

　　Log(n)基本上接近常量，比如说宇宙中原子的个数为10^69,取log后（10为底的情况）也只有69了，所以如果某个算法是log(n)的复杂度，那么这个算法是相当好的了。

　　静态查找表一般用数组实现，而动态查找表一般用树实现。查找表的实现还有键树，trie树，hash表等。

　　BST查找一定要从根节点开始，且BST的插入，查找算法一般都要用递归算法实现。可以从2-3树过渡到红黑树（红黑树的本质就是2-3-4树，比2-3树稍微复杂一点），2-3树是指每个节点的分支可以有2个或者3个。

　　红黑树中的红节点都对应于2-3-4树中大节点（指该节点内可能有2个或者3个数据）中的内部节点。

　　红黑树的查找性能和AVL相对，稍弱一点，但是实践表明，红黑树的插入过程中所需要进行的节点旋转次数比AVL树的要小。

　　2-3-4树是一颗B树，属于外部查找树。

　　红黑树的插入：

　　按照插入节点的值从红黑树的根节点依次往下插入。如果碰到其path上的节点左右节点都是红色的，则需要进行节点的颜色变换，颜色变换后如果出现了2个连续的红色节点，则需要进行旋转，旋转过程中当然也会有颜色变换。 直到找到需要插入的位置将其插入，因为插入的节点只能是红色的，所以又可能引起2个连续的红色节点，这时候仍然需要使用上面的规则进行调整。

　　红黑树的类实现code如下：

RedBlackTree.h:

#ifndef RED_BLACK_TREE_H_
#define RED_BLACK_TREE_H_

#include "Wrapper.h"

// Red-black tree class.
//
// CONSTRUCTION: with negative infinity object
//
// ******************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
// bool isEmpty( )        --> Return true if empty; else false
// void makeEmpty( )      --> Remove all items
// ******************ERRORS********************************
// Throws exceptions as warranted.

template <class Comparable>
class RedBlackTree;

template <class Comparable>
class RedBlackNode;

template <class Comparable>
class RedBlackTree
{
  public:
    RedBlackTree( const Comparable & negInf );
    RedBlackTree( const RedBlackTree & rhs );
    ~RedBlackTree( );

    Cref<Comparable> findMin( ) const;
    Cref<Comparable> findMax( ) const;
    Cref<Comparable> find( const Comparable & x ) const;
    bool isEmpty( ) const;

    void makeEmpty( );
    void insert( const Comparable & x );
    void remove( const Comparable & x );

    enum { RED, BLACK };

    const RedBlackTree & operator=( const RedBlackTree & rhs );

    typedef RedBlackNode<Comparable> Node;

  private:
    Node *header;   // The tree header (contains negInf)
    Node *nullNode;

      // Used in insert routine and its helpers (logically static)
    Node *current;
    Node *parent;
    Node *grand;
    Node *great;

      // Usual recursive stuff
    void reclaimMemory( Node *t ) const;
    RedBlackNode<Comparable> * clone( Node * t ) const;

      // Red-black tree manipulations
    void handleReorient( const Comparable & item );
    RedBlackNode<Comparable> * rotate( const Comparable & item,
                                Node *parent ) const;
    void rotateWithLeftChild( Node * & k2 ) const;
    void rotateWithRightChild( Node * & k1 ) const;
};

template <class Comparable>
class RedBlackNode
{
    Comparable    element;
    RedBlackNode *left;
    RedBlackNode *right;
    int           color;

    RedBlackNode( const Comparable & theElement = Comparable( ),
                      RedBlackNode *lt = NULL, RedBlackNode *rt = NULL,
                      int c = RedBlackTree<Comparable>::BLACK )
      : element( theElement ), left( lt ), right( rt ), color( c ) { }
    friend class RedBlackTree<Comparable>;
};

#include "RedBlackTree.cpp"
#endif

RedBlackTree.cpp:

#include "RedBlackTree.h"
#include "Except.h"

// Construct the tree.
// negInf is a value less than or equal to all others.
template <class Comparable>
RedBlackTree<Comparable>::RedBlackTree( const Comparable & negInf )
{
    nullNode    = new Node;//空节点
    nullNode->left = nullNode->right = nullNode;
    header      = new Node( negInf );//头节点，指向自己
    header->left = header->right = nullNode;
}

// Copy constructor.
template <class Comparable>
RedBlackTree<Comparable>::RedBlackTree( const RedBlackTree<Comparable> & rhs )
{
    nullNode    = new Node;
    nullNode->left = nullNode->right = nullNode;
    header      = new Node( rhs.header->element );//只用rhs树中的头节点内容构造自己的头节点
    header->left = header->right = nullNode;
    *this = rhs;
}

// Destroy the tree.
template <class Comparable>
RedBlackTree<Comparable>::~RedBlackTree( )
{
    makeEmpty( );
    delete nullNode;
    delete header;
}

// Insert item x into the tree.
// Throws DuplicateItemException if x is already present.
template <class Comparable>
void RedBlackTree<Comparable>::insert( const Comparable & x )
{
    current = parent = grand = header;//一开始都定义为头节点
    nullNode->element = x;

    while( current->element != x )//一般情况下刚调用该函数时这个whlie条件是满足的，因为此时的current->element为无穷小
    {
        great = grand; grand = parent; parent = current;//全部更新
        current = x < current->element ? current->left : current->right;

          // Check if two red children; fix if so
        if( current->left->color == RED && current->right->color == RED )//此时等价于2-3-4树中的4节点，因此需要将中间的节点往父节点方向上长
            handleReorient( x );//往上生长节点，包括旋转和颜色变换
    }

      // Insertion fails if already present
    if( current != nullNode )
        throw DuplicateItemException( );
    current = new Node( x, nullNode, nullNode );//其实current永远是需要查找的下一个，有点先行的味道

      // Attach to parent
    if( x < parent->element )
        parent->left = current;
    else
        parent->right = current;
    handleReorient( x );
}

// Remove item x from the tree.
// Not implemented in this version.
template <class Comparable>
void RedBlackTree<Comparable>::remove( const Comparable & x )
{
    cout << "Sorry, remove unimplemented; " << x <<
         " still present" << endl;
}

// Find the smallest item  the tree.
// Return the smallest item wrapped in a Cref object.
template <class Comparable>
Cref<Comparable> RedBlackTree<Comparable>::findMin( ) const
{
    if( isEmpty( ) )
        return Cref<Comparable>( );

    Node *itr = header->right;

    while( itr->left != nullNode )
        itr = itr->left;

    return Cref<Comparable>( itr->element );
}

// Find the largest item in the tree.
// Return the largest item wrapped in a Cref object.
template <class Comparable>
Cref<Comparable> RedBlackTree<Comparable>::findMax( ) const
{
    if( isEmpty( ) )
        return Cref<Comparable>( );

    Node *itr = header->right;

    while( itr->right != nullNode )
        itr = itr->right;

    return Cref<Comparable>( itr->element );
}

// Find item x in the tree.
// Return the matching item wrapped in a Cref object.
template <class Comparable>
Cref<Comparable> RedBlackTree<Comparable>::find( const Comparable & x ) const
{
    nullNode->element = x;
    Node *curr = header->right;

    for( ; ; )
    {
        if( x < curr->element )
            curr = curr->left;
        else if( curr->element < x )
            curr = curr->right;
        else if( curr != nullNode )
            return Cref<Comparable>( curr->element );
        else
            return Cref<Comparable>( );
    }
}

// Make the tree logically empty.
template <class Comparable>
void RedBlackTree<Comparable>::makeEmpty( )
{
    reclaimMemory( header->right );
    header->right = nullNode;
}

// Test if the tree is logically empty.
// Return true if empty, false otherwise.
template <class Comparable>
bool RedBlackTree<Comparable>::isEmpty( ) const
{
    return header->right == nullNode;
}

// Deep copy.
template <class Comparable>
const RedBlackTree<Comparable> &
RedBlackTree<Comparable>::operator=( const RedBlackTree<Comparable> & rhs )
{
    if( this != &rhs )
    {
        makeEmpty( );
        header->right = clone( rhs.header->right );
    }

    return *this;
}

// Internal method to clone subtree.
template <class Comparable>
RedBlackNode<Comparable> *
RedBlackTree<Comparable>::clone( Node * t ) const
{
    if( t == t->left )  // Cannot test against nullNode!!!
        return nullNode;
    else
        return new RedBlackNode<Comparable>( t->element, clone( t->left ),
                                       clone( t->right ), t->color );
}

// Internal routine that is called during an insertion
// if a node has two red children. Performs flip and rotations.
// item is the item being inserted.
template <class Comparable>
void RedBlackTree<Comparable>::handleReorient( const Comparable & item )
{
        // Do the color flip
    current->color = RED;
    current->left->color = BLACK;//空节点也被认为是黑色的
    current->right->color = BLACK;

    if( parent->color == RED )      // Have to rotate
    {
        grand->color = RED;
        if( item < grand->element != item < parent->element )//这个条件表示item是grand的内子孙，因此需要2次调整
            parent = rotate( item, grand ); // Start dbl rotate
        current = rotate( item, great );
        current->color = BLACK;
    }
    header->right->color = BLACK;   // Make root black，head其实是根节点
}

// Internal routine that performs a single or double rotation.
// Because the result is attached to the parent, there are four cases.
// Called by handleReorient.
// item is the item in handleReorient.
// parent is the parent of the root of the rotated subtree.
// Return the root of the rotated subtree.
template <class Comparable>
RedBlackNode<Comparable> *
RedBlackTree<Comparable>::rotate( const Comparable & item,
                      Node *theParent ) const
{
    if( item < theParent->element )
    {
        item < theParent->left->element ?
            rotateWithLeftChild( theParent->left )  :  // LL
            rotateWithRightChild( theParent->left ) ;  // LR
        return theParent->left;
    }
    else
    {
        item < theParent->right->element ?
            rotateWithLeftChild( theParent->right ) :  // RL
            rotateWithRightChild( theParent->right );  // RR
        return theParent->right;
    }
}

// Rotate binary tree node with left child.
template <class Comparable>
void RedBlackTree<Comparable>::
rotateWithLeftChild( Node * & k2 ) const
{
    Node *k1 = k2->left;
    k2->left = k1->right;
    k1->right = k2;
    k2 = k1;
}

// Rotate binary tree node with right child.
template <class Comparable>
void RedBlackTree<Comparable>::
rotateWithRightChild( Node * & k1 ) const
{
    Node *k2 = k1->right;
    k1->right = k2->left;
    k2->left = k1;
    k1 = k2;
}

// Internal method to reclaim internal nodes in subtree t.
template <class Comparable>
void RedBlackTree<Comparable>::reclaimMemory( Node *t ) const
{
    if( t != t->left )
    {
        reclaimMemory( t->left );
        reclaimMemory( t->right );
        delete t;
    }
}

　　参考资料：

　　data structures and algorithm analysis in c++ (second edition)，mark allen Weiss.

算法设计和数据结构学习_4(《数据结构和问题求解》part4笔记)