问题六十四：怎么用C++实现二叉查找树（binary search tree）及其相关操作

64.0 概述

什么是二叉查找树（binary search tree）？

二叉查找树（binary search tree）又叫二叉排序树（binary ordered tree）。

对于任意二叉查找树，要么是一棵空树，要么具有如下性质：

若它的左子树不为空，则左子树上所有结点的值都小于它的根结点的值；

若它的右子树不为空，则右子树上所有结点的值都大于它的根结点的值；

它的左、右子树也分别为二叉查找树；

中序遍历二叉查找树可得到一个关键字的有序序列，一个无序序列可以通过构造一棵二叉查找树变成一个有序序列，构造树的过程即为对无序序列进行排序的过程；

 

我们这里用到的二叉查找树的数据部分的数据类型为整型，如下代码为int定义一个别名“ItemType”：

typedef int ItemType;

如下代码是定义二叉查找树结点。每一个结点包含：数据部分，左孩子指针，右孩子指针。

struct TreeNode{    ItemType info;    TreeNode* left;    TreeNode* right;};

 

如下定义二叉查找树类：定义伊始，是一棵空树。

class TreeType {// binary search tree.    public:        TreeType() {            root = NULL;        }//在这个位置添加二叉查找树的各种操作方法的声明；        TreeNode *root;};

 

新建二叉查找树对象：

         TreeType binarySearchTree;

64.1 建立二叉查找树

下面，我们在已经定义的空的二叉查找树的基础上，建立有数据的树。

 

我们的做法是：将如下数组中的15个数据，依次建立二叉查找树。（相当于将数组中数据依次插入之前新建的空的二叉查找树）

         ItemType infoArray[15] = {20, 12, 2, 14, 15, 35, 6, 45, 25, 23, 1, 24, 16, 57, 28};

根据二叉查找树的定义和性质，如上数组中的15个数据对应的二叉查找树如下：

算法：

1，第一个被插入的数据（20）成为二叉查找树的根结点；

2，有了根结点之后，后面的数据被插入的位置取决于其与根结点值的大小关系：比根结点的值小，则在根结点的左子树中；比根结点的值大，则在根结点的右子树中。所以，关键是找到被插入数据未来的父结点。这样的父结点需要满足两个条件：其一，其值满足如上的大小关系；其二，其对应的左孩子结点或者右孩子结点（左、右由值的大小关系决定）的位置是空的。

 

对应C++代码如下：

----------------------------------------------TreeType.h------------------------------------------

TreeType.h

        void CreateTree(ItemType *itemArray, int itemNum);

----------------------------------------------TreeType.cpp------------------------------------------

TreeType.cpp

void TreeType::CreateTree(ItemType *itemArray, int itemNum) {     for(int i=0; i<itemNum; i++) {         TreeNode *newnode = new TreeNode;         newnode->info = itemArray[i];         newnode->left = NULL;         newnode->right = NULL;         if(root == NULL) {             root = newnode;             std::cout << itemArray[i] << " will be the root" << endl;         }         else {             TreeNode *parent;             TreeNode *temp = root;             while(temp != NULL) {//这个循环是找未来的父结点。                 parent = temp;                 if((temp->info) > itemArray[i]) {                     temp = temp->left;                 }                 else {                     temp = temp->right;                 }             }             if((parent->info) > itemArray[i]) {//这个if……else……是根据被插入值和父结点的大小关系，确定被插入的位置                 parent->left = newnode;                 std::cout << itemArray[i] << " will be " << parent->info << "'s left child" << endl;             }             else {                 parent->right = newnode;                 std::cout << itemArray[i] << " will be " << parent->info << "'s right child" << endl;             }         }    }}

 

----------------------------------------------main.cpp------------------------------------------

main.cpp

         TreeType binarySearchTree;         ItemType infoArray[15] = {20, 12, 2, 14, 15, 35, 6, 45, 25, 23, 1, 24, 16, 57, 28};         std::cout << "CreateTree: " << endl;         binarySearchTree.CreateTree(infoArray, 15);         std::cout << endl;

 

输出结果如下：

64.2 遍历二叉查找树

我们在“64.1”中建立了一棵二叉查找树，现在我们想看看这个树中每个结点的内容。我们需要遍历这个二叉查找树。

有三种遍历二叉查找树的方式：（根）前序遍历、（根）中序遍历、（根）后序遍历

如上二叉查找树的（根）前序遍历的结果应该是：20， 12， 2， 1， 6， 14， 15， 16， 35， 25， 23， 24， 28， 45， 57

如上二叉查找树的（根）前序遍历的结果应该是：1， 2， 6， 12， 14， 15， 16， 20， 23， 24， 25， 28， 35， 45， 57

如上二叉查找树的（根）前序遍历的结果应该是：1， 6， 2， 16， 15， 14， 12， 24， 23， 28， 25， 57， 45， 35， 20

 

我们采用递归的方法来完成遍历。

 

C++代码如下：

----------------------------------------------TreeType.h------------------------------------------

TreeType.h

        void PreorderTraverse(TreeNode *rootNode);//（根）前序遍历        void InorderTraverse(TreeNode *rootNode); //（根）中序遍历        void PostorderTraverse(TreeNode *rootNode); //（根）后序遍历

----------------------------------------------TreeType.cpp------------------------------------------

TreeType.cpp

void TreeType::PreorderTraverse(TreeNode *rootNode) {     if(rootNode!=NULL)     {         std::cout << rootNode->info << " ";//遍历根结点         PreorderTraverse(rootNode->left);//遍历左子树         PreorderTraverse(rootNode->right);//遍历右子树     } } void TreeType::InorderTraverse(TreeNode *rootNode) {     if(rootNode!=NULL)     {         InorderTraverse(rootNode->left); //遍历左子树         std::cout << rootNode->info << " ";//遍历根结点         InorderTraverse(rootNode->right); //遍历右子树     } } void TreeType::PostorderTraverse(TreeNode *rootNode) {     if(rootNode!=NULL)     {         PostorderTraverse(rootNode->left); //遍历左子树         PostorderTraverse(rootNode->right); //遍历右子树         std::cout << rootNode->info << " ";//遍历根结点     } }

 

----------------------------------------------main.cpp------------------------------------------

main.cpp 

         std::cout << "PreorderTraverse: " << endl;         binarySearchTree.PreorderTraverse(binarySearchTree.root);         std::cout << endl;         std::cout << "InorderTraverse: " << endl;         binarySearchTree.InorderTraverse(binarySearchTree.root);         std::cout << endl;         std::cout << "PostorderTraverse: " << endl;         binarySearchTree.PostorderTraverse(binarySearchTree.root);         std::cout << endl;         std::cout << endl;

 

输出结果如下：

64.3 向二叉查找树中插入若干个结点

这个方式和“64.1 建立二叉查找树”基本完全相同，此处不赘述，贴出代码如下：

----------------------------------------------TreeType.h------------------------------------------

TreeType.h

        void InsertItems(ItemType *itemArray, int itemNum);

----------------------------------------------TreeType.cpp------------------------------------------

TreeType.cpp

void TreeType::InsertItems(ItemType *itemArray, int itemNum) {     for(int i=0; i<itemNum; i++) {         TreeNode *newnode = new TreeNode;         newnode->info = itemArray[i];         newnode->left = NULL;         newnode->right = NULL;         if(root == NULL) {             root = newnode;             std::cout << itemArray[i] << " will be the root" << endl;         }         else {             TreeNode *parent;             TreeNode *temp = root;             while(temp != NULL) {                 parent = temp;                 if((temp->info) > itemArray[i]) {                     temp = temp->left;                 }                 else if((temp->info) == itemArray[i]) {                    break;                 }                 else {                     temp = temp->right;                 }             }             if((parent->info) > itemArray[i]) {                 parent->left = newnode;                 std::cout << itemArray[i] << " will be " << parent->info << "'s left child" << endl;             }             else if((parent->info) == itemArray[i]) {                 std::cout << "there has been a " << itemArray[i] << " in the tree" << endl;             }             else {                 parent->right = newnode;                 std::cout << itemArray[i] << " will be " << parent->info << "'s right child" << endl;             }         }    }}

----------------------------------------------main.cpp------------------------------------------

main.cpp

/*先插入4个数据，其中的“28”已经在树中，所以实际插入的是3个数据：40、32、42。然后将整棵树按照三种方式分别进行遍历*/

         ItemType insertItems[4] = {40, 32, 42, 28};         std::cout << "InsertItems: ";         for (int i=0; i<4; i++) {            std::cout << insertItems[i] << " ";         }         std::cout << endl;         binarySearchTree.InsertItems(insertItems, 4);         std::cout << endl;         std::cout << "PreorderTraverse: " << endl;         binarySearchTree.PreorderTraverse(binarySearchTree.root);         std::cout << endl;         std::cout << "InorderTraverse: " << endl;         binarySearchTree.InorderTraverse(binarySearchTree.root);         std::cout << endl;         std::cout << "PostorderTraverse: " << endl;         binarySearchTree.PostorderTraverse(binarySearchTree.root);         std::cout << endl;

                                                                            

输出结果如下：

对应二叉查找树的示意图如下：

64.4 查找二叉查找树中某个结点

我们采用递归的方法来完成某个结点的查找。

算法：

先查找根结点，若根结点是要找的结点，直接输出根结点内容；

若根结点不是要找的结点，按照递归的方法分别查找左子树和右子树；

 

C++代码如下：

 

----------------------------------------------TreeType.h------------------------------------------

TreeType.h

        bool FindItem(TreeNode *rootNode, ItemType item);

----------------------------------------------TreeType.cpp------------------------------------------

TreeType.cpp

bool TreeType::FindItem(TreeNode *rootNode, ItemType item) {    if (rootNode == NULL) {        return  false;    }    else {        if ((rootNode->info) == item) {            std::cout << "found." << endl;            if ((rootNode->left) != NULL) {                std::cout << "its left child is " << rootNode->left->info << " ." << endl;            }            if ((rootNode->right) != NULL) {                std::cout << "its right child is " << rootNode->right->info << " ." << endl;            }            return true;        }        else {            if (FindItem(rootNode->left, item)) {                return true;            }            if (FindItem(rootNode->right, item)) {                return true;            }            return false;        }    }}ItemType TreeType::MinValue(TreeNode *rootNode) {     if(rootNode!=NULL)     {         if ((rootNode->left) != NULL) {            MinValue(rootNode->left);         }         else {            return rootNode->info;         }     }}

 

----------------------------------------------main.cpp------------------------------------------

main.cpp

         std::cout << endl;         ItemType item = 35;         bool found;         std::cout << "FindItem: " << item << endl;         found = binarySearchTree.FindItem(binarySearchTree.root, item);         std::cout << "found: " << found << endl;

 

输出结果如下：

64.5 查找二叉查找树中的最大值和最小值

我们知道，二叉查找树是有序序列，而且左子树中所有的值都比根结点的值小，右子树中所有的值都比根结点的值大。所以，最大值是最右边的那个值，最小值是最左边的那个值。进一步分析：最大值为中序遍历的最后一个值，最小值为中序遍历大的第一个值。

 

C++代码如下：

 

----------------------------------------------TreeType.h------------------------------------------

TreeType.h

        ItemType MinValue(TreeNode *rootNode);        ItemType MaxValue(TreeNode *rootNode);

----------------------------------------------TreeType.cpp------------------------------------------

TreeType.cpp

ItemType TreeType::MaxValue(TreeNode *rootNode) {     if(rootNode!=NULL)     {         if ((rootNode->right) != NULL) {            MaxValue(rootNode->right);         }         else {            return rootNode->info;         }     }}

----------------------------------------------main.cpp------------------------------------------

main.cpp

         std::cout << endl;         std::cout << "MinValue: " << endl;         ItemType minItem = binarySearchTree.MinValue(binarySearchTree.root);         std::cout << "minItem: " << minItem << endl;         std::cout << endl;         std::cout << "MaxValue: " << endl;         ItemType maxItem = binarySearchTree.MaxValue(binarySearchTree.root);         std::cout << "maxItem: " << maxItem << endl;

 

输出结果如下：

64.6 求二叉查找树的总的结点数

我们采用递归的方法来求二叉查找树的总的结点数。

算法：

总的结点数=左子树的结点数+右子树的结点数+1（根结点）

 

C++代码如下：

 

----------------------------------------------TreeType.h------------------------------------------

TreeType.h

        int NodesNum(TreeNode *rootNode);

----------------------------------------------TreeType.cpp------------------------------------------

TreeType.cpp

int TreeType::NodesNum(TreeNode *rootNode) {     if(rootNode!=NULL)     {        return (NodesNum(rootNode->left) + NodesNum(rootNode->right) + 1);     }     else {        return 0;     }}

----------------------------------------------main.cpp------------------------------------------

main.cpp

         std::cout << endl;         std::cout << "NodesNum: " << endl;         int num = binarySearchTree.NodesNum(binarySearchTree.root);         std::cout << "NodesNum: " << num << endl;

输出结果如下：

64.7 求二叉查找树中叶子结点的个数及遍历叶子结点

叶子结点，即没有左孩子和右孩子的结点。如上二叉查找树的叶子结点如下图蓝色圈示意：

 

C++代码实现如下：

 

----------------------------------------------TreeType.h------------------------------------------

TreeType.h

        void LeavesNum(TreeNode *rootNode, int &num);        void PreorderTraverseLeaves(TreeNode *rootNode);        void InorderTraverseLeaves(TreeNode *rootNode);        void PostorderTraverseLeaves(TreeNode *rootNode);

 

----------------------------------------------TreeType.cpp------------------------------------------

TreeType.cpp

void TreeType::LeavesNum(TreeNode *rootNode, int &num) {     if(rootNode!=NULL)     {        if (((rootNode->left) == NULL) && ((rootNode->right) == NULL)) {            num ++;        }        else {            LeavesNum(rootNode->left, num);            LeavesNum(rootNode->right, num);        }     }}void TreeType::PreorderTraverseLeaves(TreeNode *rootNode) {    if(rootNode!=NULL) {        if ((rootNode->left == NULL) && (rootNode->right == NULL)) {            std::cout << rootNode->info << " ";        }        PreorderTraverseLeaves(rootNode->left);        PreorderTraverseLeaves(rootNode->right);    }}void TreeType::InorderTraverseLeaves(TreeNode *rootNode) {    if(rootNode!=NULL) {        InorderTraverseLeaves(rootNode->left);        if ((rootNode->left == NULL) && (rootNode->right == NULL)) {            std::cout << rootNode->info << " ";        }        InorderTraverseLeaves(rootNode->right);    }}void TreeType::PostorderTraverseLeaves(TreeNode *rootNode) {    if(rootNode!=NULL)    {        PostorderTraverseLeaves(rootNode->left);        PostorderTraverseLeaves(rootNode->right);        if ((rootNode->left == NULL) && (rootNode->right == NULL)) {            std::cout << rootNode->info << " ";        }    }}

 

----------------------------------------------main.cpp------------------------------------------

main.cpp

         std::cout << endl;         std::cout << "LeavesNum: " << endl;         int lnum = 0;         binarySearchTree.LeavesNum(binarySearchTree.root, lnum);         std::cout << "LeavesNum: " << lnum << endl;         std::cout << "PreorderTraverseLeaves: " << endl;         binarySearchTree.PreorderTraverseLeaves(binarySearchTree.root);         std::cout << endl;         std::cout << "InorderTraverseLeaves: " << endl;         binarySearchTree.InorderTraverseLeaves(binarySearchTree.root);         std::cout << endl;         std::cout << "PostorderTraverseLeaves: " << endl;         binarySearchTree.PostorderTraverseLeaves(binarySearchTree.root);         std::cout << endl;

 

输出结果如下：

 

64.8 删除二叉查找树中某个结点

删除某个结点，步骤如下：

1，在二叉查找树中找到该结点；

2，将该结点的父结点对应的左孩子或者右孩子的指针设为空；

3，删除以该结点为根结点的子树；

 

比如，若删除结点“45”，则是下图中蓝色圈内的4个结点。

 

C++代码如下：

 

----------------------------------------------TreeType.h------------------------------------------

TreeType.h

        bool DeleteItem(TreeNode *rootNode, ItemType item);

 

----------------------------------------------TreeType.cpp------------------------------------------

TreeType.cpp

void DeleteTree(TreeNode *rootNode) {    if (rootNode == NULL) {    }    else {        TreeNode *temp_left = rootNode->left;        TreeNode *temp_right = rootNode->right;        delete rootNode;        if (temp_left != NULL) {            DeleteTree(temp_left);        }        if (temp_right != NULL) {            DeleteTree(temp_right);        }    } }bool TreeType::DeleteItem(TreeNode *rootNode, ItemType item) {    if (rootNode == NULL) {        return  false;    }    else {        if (rootNode->left != NULL) {            if (rootNode->left->info == item) {                DeleteTree(rootNode->left);                rootNode->left = NULL;                return true;            }        }        if (rootNode->right != NULL) {            if (rootNode->right->info == item) {                DeleteTree(rootNode->right);                rootNode->right = NULL;                return true;            }        }        if (DeleteItem(rootNode->left, item)) {            return true;        }        if (DeleteItem(rootNode->right, item)) {            return true;        }        else {            return false;        }    } }

----------------------------------------------main.cpp------------------------------------------

main.cpp

/*先删除结点“45”（即以该结点为根结点的子树）。然后在遍历完成删除动作后剩余的树，同时计算剩余结点数，和剩余叶子结点数，再遍历叶子结点*/

         std::cout << endl;         item = 45;         std::cout << "DeleteItem: " << item << endl;         bool deleted = binarySearchTree.DeleteItem(binarySearchTree.root, item);         std::cout << "deleted: " << deleted << endl;         std::cout << endl;         std::cout << "PreorderTraverse: " << endl;         binarySearchTree.PreorderTraverse(binarySearchTree.root);         std::cout << endl;         std::cout << "InorderTraverse: " << endl;         binarySearchTree.InorderTraverse(binarySearchTree.root);         std::cout << endl;         std::cout << "PostorderTraverse: " << endl;         binarySearchTree.PostorderTraverse(binarySearchTree.root);         std::cout << endl;         std::cout << endl;         std::cout << "NodesNum: " << endl;         num = binarySearchTree.NodesNum(binarySearchTree.root);         std::cout << "NodesNum: " << num << endl;         std::cout << endl;         std::cout << "LeavesNum: " << endl;         lnum = 0;         binarySearchTree.LeavesNum(binarySearchTree.root, lnum);         std::cout << "LeavesNum: " << lnum << endl;         std::cout << "PreorderTraverseLeaves: " << endl;         binarySearchTree.PreorderTraverseLeaves(binarySearchTree.root);         std::cout << endl;         std::cout << "InorderTraverseLeaves: " << endl;         binarySearchTree.InorderTraverseLeaves(binarySearchTree.root);         std::cout << endl;         std::cout << "PostorderTraverseLeaves: " << endl;         binarySearchTree.PostorderTraverseLeaves(binarySearchTree.root);         std::cout << endl;

 

输出结果如下：

64.9 C++代码汇总

----------------------------------------------TreeType.h------------------------------------------

TreeType.h

#ifndef TREETYPE_H#define TREETYPE_H#include <iostream>using namespace std;typedef int ItemType;struct TreeNode{    ItemType info;    TreeNode* left;    TreeNode* right;};class TreeType {// binary search tree.    public:        TreeType() {            root = NULL;        }        void CreateTree(ItemType *itemArray, int itemNum);        void PreorderTraverse(TreeNode *rootNode);        void InorderTraverse(TreeNode *rootNode);        void PostorderTraverse(TreeNode *rootNode);        void InsertItems(ItemType *itemArray, int itemNum);        bool FindItem(TreeNode *rootNode, ItemType item);        ItemType MinValue(TreeNode *rootNode);        ItemType MaxValue(TreeNode *rootNode);        int NodesNum(TreeNode *rootNode);        void LeavesNum(TreeNode *rootNode, int &num);        void PreorderTraverseLeaves(TreeNode *rootNode);        void InorderTraverseLeaves(TreeNode *rootNode);        void PostorderTraverseLeaves(TreeNode *rootNode);        bool DeleteItem(TreeNode *rootNode, ItemType item);        TreeNode *root;};#endif // TREETYPE_H

----------------------------------------------TreeType.cpp------------------------------------------

TreeType.cpp

#include "TreeType.h"void TreeType::CreateTree(ItemType *itemArray, int itemNum) {     for(int i=0; i<itemNum; i++) {         TreeNode *newnode = new TreeNode;         newnode->info = itemArray[i];         newnode->left = NULL;         newnode->right = NULL;         if(root == NULL) {             root = newnode;             std::cout << itemArray[i] << " will be the root" << endl;         }         else {             TreeNode *parent;             TreeNode *temp = root;             while(temp != NULL) {                 parent = temp;                 if((temp->info) > itemArray[i]) {                     temp = temp->left;                 }                 else {                     temp = temp->right;                 }             }             if((parent->info) > itemArray[i]) {                 parent->left = newnode;                 std::cout << itemArray[i] << " will be " << parent->info << "'s left child" << endl;             }             else {                 parent->right = newnode;                 std::cout << itemArray[i] << " will be " << parent->info << "'s right child" << endl;             }         }    }} void TreeType::PreorderTraverse(TreeNode *rootNode) {     if(rootNode!=NULL)     {         std::cout << rootNode->info << " ";         PreorderTraverse(rootNode->left);         PreorderTraverse(rootNode->right);     } } void TreeType::InorderTraverse(TreeNode *rootNode) {     if(rootNode!=NULL)     {         InorderTraverse(rootNode->left);         std::cout << rootNode->info << " ";         InorderTraverse(rootNode->right);     } } void TreeType::PostorderTraverse(TreeNode *rootNode) {     if(rootNode!=NULL)     {         PostorderTraverse(rootNode->left);         PostorderTraverse(rootNode->right);         std::cout << rootNode->info << " ";     } }void TreeType::InsertItems(ItemType *itemArray, int itemNum) {     for(int i=0; i<itemNum; i++) {         TreeNode *newnode = new TreeNode;         newnode->info = itemArray[i];         newnode->left = NULL;         newnode->right = NULL;         if(root == NULL) {             root = newnode;             std::cout << itemArray[i] << " will be the root" << endl;         }         else {             TreeNode *parent;             TreeNode *temp = root;             while(temp != NULL) {                 parent = temp;                 if((temp->info) > itemArray[i]) {                     temp = temp->left;                 }                 else if((temp->info) == itemArray[i]) {                    break;                 }                 else {                     temp = temp->right;                 }             }             if((parent->info) > itemArray[i]) {                 parent->left = newnode;                 std::cout << itemArray[i] << " will be " << parent->info << "'s left child" << endl;             }             else if((parent->info) == itemArray[i]) {                 std::cout << "there has been a " << itemArray[i] << " in the tree" << endl;             }             else {                 parent->right = newnode;                 std::cout << itemArray[i] << " will be " << parent->info << "'s right child" << endl;             }         }    }}bool TreeType::FindItem(TreeNode *rootNode, ItemType item) {    if (rootNode == NULL) {        return  false;    }    else {        if ((rootNode->info) == item) {            std::cout << "found." << endl;            if ((rootNode->left) != NULL) {                std::cout << "its left child is " << rootNode->left->info << " ." << endl;            }            if ((rootNode->right) != NULL) {                std::cout << "its right child is " << rootNode->right->info << " ." << endl;            }            return true;        }        else {            if (FindItem(rootNode->left, item)) {                return true;            }            if (FindItem(rootNode->right, item)) {                return true;            }            return false;        }    }}ItemType TreeType::MinValue(TreeNode *rootNode) {     if(rootNode!=NULL)     {         if ((rootNode->left) != NULL) {            MinValue(rootNode->left);         }         else {            return rootNode->info;         }     }}ItemType TreeType::MaxValue(TreeNode *rootNode) {     if(rootNode!=NULL)     {         if ((rootNode->right) != NULL) {            MaxValue(rootNode->right);         }         else {            return rootNode->info;         }     }}int TreeType::NodesNum(TreeNode *rootNode) {     if(rootNode!=NULL)     {        return (NodesNum(rootNode->left) + NodesNum(rootNode->right) + 1);     }     else {        return 0;     }}void TreeType::LeavesNum(TreeNode *rootNode, int &num) {     if(rootNode!=NULL)     {        if (((rootNode->left) == NULL) && ((rootNode->right) == NULL)) {            num ++;        }        else {            LeavesNum(rootNode->left, num);            LeavesNum(rootNode->right, num);        }     }}void TreeType::PreorderTraverseLeaves(TreeNode *rootNode) {    if(rootNode!=NULL) {        if ((rootNode->left == NULL) && (rootNode->right == NULL)) {            std::cout << rootNode->info << " ";        }        PreorderTraverseLeaves(rootNode->left);        PreorderTraverseLeaves(rootNode->right);    }}void TreeType::InorderTraverseLeaves(TreeNode *rootNode) {    if(rootNode!=NULL) {        InorderTraverseLeaves(rootNode->left);        if ((rootNode->left == NULL) && (rootNode->right == NULL)) {            std::cout << rootNode->info << " ";        }        InorderTraverseLeaves(rootNode->right);    }}void TreeType::PostorderTraverseLeaves(TreeNode *rootNode) {    if(rootNode!=NULL)    {        PostorderTraverseLeaves(rootNode->left);        PostorderTraverseLeaves(rootNode->right);        if ((rootNode->left == NULL) && (rootNode->right == NULL)) {            std::cout << rootNode->info << " ";        }    }}void DeleteTree(TreeNode *rootNode) {    if (rootNode == NULL) {    }    else {        TreeNode *temp_left = rootNode->left;        TreeNode *temp_right = rootNode->right;        delete rootNode;        if (temp_left != NULL) {            DeleteTree(temp_left);        }        if (temp_right != NULL) {            DeleteTree(temp_right);        }    } } bool TreeType::DeleteItem(TreeNode *rootNode, ItemType item) {    if (rootNode == NULL) {        return  false;    }    else {        if (rootNode->left != NULL) {            if (rootNode->left->info == item) {                DeleteTree(rootNode->left);                rootNode->left = NULL;                return true;            }        }        if (rootNode->right != NULL) {            if (rootNode->right->info == item) {                DeleteTree(rootNode->right);                rootNode->right = NULL;                return true;            }        }        if (DeleteItem(rootNode->left, item)) {            return true;        }        if (DeleteItem(rootNode->right, item)) {            return true;        }        else {            return false;        }    } }

 

----------------------------------------------main.cpp------------------------------------------

main.cpp

    #include <iostream>    #include <fstream>    #include <limits>    #include <stdlib.h>    #include <TreeType.h>    using namespace std;    int main()    {         TreeType binarySearchTree;         ItemType infoArray[15] = {20, 12, 2, 14, 15, 35, 6, 45, 25, 23, 1, 24, 16, 57, 28};         std::cout << "CreateTree: " << endl;         binarySearchTree.CreateTree(infoArray, 15);         std::cout << endl;         std::cout << "PreorderTraverse: " << endl;         binarySearchTree.PreorderTraverse(binarySearchTree.root);         std::cout << endl;         std::cout << "InorderTraverse: " << endl;         binarySearchTree.InorderTraverse(binarySearchTree.root);         std::cout << endl;         std::cout << "PostorderTraverse: " << endl;         binarySearchTree.PostorderTraverse(binarySearchTree.root);         std::cout << endl;         std::cout << endl;         ItemType insertItems[4] = {40, 32, 42, 28};         std::cout << "InsertItems: ";         for (int i=0; i<4; i++) {            std::cout << insertItems[i] << " ";         }         std::cout << endl;         binarySearchTree.InsertItems(insertItems, 4);         std::cout << endl;         std::cout << "PreorderTraverse: " << endl;         binarySearchTree.PreorderTraverse(binarySearchTree.root);         std::cout << endl;         std::cout << "InorderTraverse: " << endl;         binarySearchTree.InorderTraverse(binarySearchTree.root);         std::cout << endl;         std::cout << "PostorderTraverse: " << endl;         binarySearchTree.PostorderTraverse(binarySearchTree.root);         std::cout << endl;         std::cout << endl;         ItemType item = 35;         bool found;         std::cout << "FindItem: " << item << endl;         found = binarySearchTree.FindItem(binarySearchTree.root, item);         std::cout << "found: " << found << endl;         std::cout << endl;         std::cout << "MinValue: " << endl;         ItemType minItem = binarySearchTree.MinValue(binarySearchTree.root);         std::cout << "minItem: " << minItem << endl;         std::cout << endl;         std::cout << "MaxValue: " << endl;         ItemType maxItem = binarySearchTree.MaxValue(binarySearchTree.root);         std::cout << "maxItem: " << maxItem << endl;         std::cout << endl;         std::cout << "NodesNum: " << endl;         int num = binarySearchTree.NodesNum(binarySearchTree.root);         std::cout << "NodesNum: " << num << endl;         std::cout << endl;         std::cout << "LeavesNum: " << endl;         int lnum = 0;         binarySearchTree.LeavesNum(binarySearchTree.root, lnum);         std::cout << "LeavesNum: " << lnum << endl;         std::cout << "PreorderTraverseLeaves: " << endl;         binarySearchTree.PreorderTraverseLeaves(binarySearchTree.root);         std::cout << endl;         std::cout << "InorderTraverseLeaves: " << endl;         binarySearchTree.InorderTraverseLeaves(binarySearchTree.root);         std::cout << endl;         std::cout << "PostorderTraverseLeaves: " << endl;         binarySearchTree.PostorderTraverseLeaves(binarySearchTree.root);         std::cout << endl;         std::cout << endl;         item = 45;         std::cout << "DeleteItem: " << item << endl;         bool deleted = binarySearchTree.DeleteItem(binarySearchTree.root, item);         std::cout << "deleted: " << deleted << endl;         std::cout << endl;         std::cout << "PreorderTraverse: " << endl;         binarySearchTree.PreorderTraverse(binarySearchTree.root);         std::cout << endl;         std::cout << "InorderTraverse: " << endl;         binarySearchTree.InorderTraverse(binarySearchTree.root);         std::cout << endl;         std::cout << "PostorderTraverse: " << endl;         binarySearchTree.PostorderTraverse(binarySearchTree.root);         std::cout << endl;         std::cout << endl;         std::cout << "NodesNum: " << endl;         num = binarySearchTree.NodesNum(binarySearchTree.root);         std::cout << "NodesNum: " << num << endl;         std::cout << endl;         std::cout << "LeavesNum: " << endl;         lnum = 0;         binarySearchTree.LeavesNum(binarySearchTree.root, lnum);         std::cout << "LeavesNum: " << lnum << endl;         std::cout << "PreorderTraverseLeaves: " << endl;         binarySearchTree.PreorderTraverseLeaves(binarySearchTree.root);         std::cout << endl;         std::cout << "InorderTraverseLeaves: " << endl;         binarySearchTree.InorderTraverseLeaves(binarySearchTree.root);         std::cout << endl;         std::cout << "PostorderTraverseLeaves: " << endl;         binarySearchTree.PostorderTraverseLeaves(binarySearchTree.root);         std::cout << endl;    }

 

输出结果如下：