【数据结构】67_二叉树的典型遍历方式

问题： 二叉树是否只有一种遍历方式 （层次遍历）？

典型的二叉树遍历方式

• 先序遍历 （Pre-Order Traversal）
• 中序遍历 （In-Order Traversal）
• 后序遍历 （Post-order Traversal）

先序遍历 （Pre-Order Traversal）

• 二叉树为空

  • 无操作，直接返回

• 二叉树不为空

  • 访问根结点中的数据元素
  • 先序遍历左子树
  • 先序遍历右子树

先序遍历功能定义

preOrderTraversal(node)
{
    if (node != NULL)
    {
        access(node->value);
        preOrderTraversal(node->left);
        preOrderTraversal(node->right);
    }
}

中序遍历 (In-Order Traversal)

• 二叉树为空

  • 无操作，直接返回

• 二叉树不为空

  • 中序遍历左子树
  • 访问根结点中的数据元素
  • 中序遍历右子树

中序遍历功能定义

inOrderTraversal(node)
{
    if (node != NULL)
    {
        inOrderTraversal(node->left);
        access(node->value);
        inOrderTraversal(node->right);
    }
}

后序遍历 (Post-Order Traversal)

二叉树为空

• 无操作，直接返回

二叉树不为空

• 后序遍历左子树
• 后序遍历右子树
• 访问根结点中的数据元素

后序遍历功能定义

postOrderTraversal(node)
{
    postOrderTraversal(node->left);
    postOrderTraversal(node->right);
    access(node->value);
}

---

问题：是否可以将二叉树的典型遍历算法集成到 BTree 中？如果可以，代码需要做怎样的改动

设计要点

• 不能与层次遍历函数冲突，必须设计新的函数接口
• 算法执行完成后，能够方便的获得遍历结果
• 遍历结果能够反映结点的先后次序

函数结构设计

• SharedPointer<Array<T>> traversal(BTTraversal order)

  • 根据参数 order 选择遍历算法（先序，中序，后序）
  • 返回值为堆中的数组对象（生命期由智能指针管理）
  • 数组元素的次序反应遍历的先后次序

典型遍历示例

SharedPointer<Array<int>> sp = NULL;

sp = tree.traversal(PreOrder);

for (int i=0; i<(*sp).length(); ++i)
{
    cout << (*sp)[i] << endl;
}

编程实验：二叉树的典型遍历方式

文件：BTree.h

#ifndef BTREE_H
#define BTREE_H

#include "Tree.h"
#include "BTreeNode.h"
#include "Exception.h"
#include "LinkQueue.h"
#include "DynamicArray.h"

namespace DTLib
{

enum BTTraversal
{
    PreOrder,
    InOrder,
    PostOrder
};

template <typename T>
class BTree : public Tree<T>
{
public:
    BTree() = default;

    bool insert(TreeNode<T> *node) override
    {
        return insert(node, ANY);
    }

    virtual bool insert(TreeNode<T> *node, BTNodePos pos)
    {
        bool ret = true;

        if (node != nullptr)
        {
            if (this->m_root == nullptr)
            {
                node->parent = nullptr;
                this->m_root = node;
            }
            else
            {
                BTreeNode<T> *np = find(node->parent);

                if (np != nullptr)
                {
                    ret = insert(dynamic_cast<BTreeNode<T>*>(node), np, pos);
                }
                else
                {
                    THROW_EXCEPTION(InvalidParameterExcetion, "Invalid parent tree node ...");
                }
            }
        }
        else
        {
            THROW_EXCEPTION(InvalidParameterExcetion, "Parameter can not be null ...");
        }

        return ret;
    }

    bool insert(const T &value, TreeNode<T> *parent) override
    {
        return insert(value, parent, ANY);
    }

    virtual bool insert(const T &value, TreeNode<T> *parent, BTNodePos pos)
    {
        bool ret = true;
        BTreeNode<T> *node = BTreeNode<T>::NewNode();

        if (node != nullptr)
        {
            node->value = value;
            node->parent = parent;

            ret = insert(node, pos);

            if (!ret)
            {
                delete node;
            }
        }
        else
        {
            THROW_EXCEPTION(NoEnoughMemoryException, "No enough memory to create node ...");
        }

        return ret;
    }

    SharedPointer<Tree<T>> remove(const T &value) override
    {
        BTree<T> *ret = nullptr;

        BTreeNode<T> *node = find(value);

        if (node != nullptr)
        {
            remove(node, ret);

            m_queue.clear();
        }
        else
        {
            THROW_EXCEPTION(InvalidParameterExcetion, "Can not find the tree node via value ...");
        }

        return ret;
    }

    SharedPointer<Tree<T>> remove(TreeNode<T> *node) override
    {
        BTree<T> *ret = nullptr;

        node = find(node);

        if (node != nullptr)
        {
            remove(dynamic_cast<BTreeNode<T>*>(node), ret);

            m_queue.clear();
        }
        else
        {
            THROW_EXCEPTION(InvalidParameterExcetion, "Parameter node is invalid ...");
        }

        return ret;
    }

    BTreeNode<T>* find(const T &value) const override
    {
        return find(root(), value);
    }

    BTreeNode<T>* find(TreeNode<T> *node) const override
    {
        return find(root(), dynamic_cast<BTreeNode<T>*>(node));
    }

    BTreeNode<T>* root() const override
    {
        return dynamic_cast<BTreeNode<T>*>(this->m_root);
    }

    int degree() const override
    {
        return degree(root());
    }

    int count() const override
    {
        return count(root());
    }

    int height() const override
    {
        return height(root());
    }

    void clear() override
    {
        free(root());

        this->m_root = nullptr;
    }

    bool begin() override
    {
        bool ret = (root() != nullptr);

        if (ret)
        {
            m_queue.clear();
            m_queue.add(root());
        }

        return ret;
    }

    bool end() override
    {
        return (m_queue.length() == 0);
    }

    bool next() override
    {
        bool ret = (m_queue.length() > 0);

        if (ret)
        {
            BTreeNode<T> *node = m_queue.front();

            m_queue.remove();

            if (node->left != nullptr)
            {
                m_queue.add(node->left);
            }

            if (node->right != nullptr)
            {
                m_queue.add(node->right);
            }
        }

        return ret;
    }

    T current() override
    {
        if (!end())
        {
            return m_queue.front()->value;
        }
        else
        {
            THROW_EXCEPTION(InvalidOpertionExcetion, "No value at current position ...");
        }
    }

    SharedPointer<DynamicArray<T>> traversal(BTTraversal order) const
    {
        DynamicArray<T> *ret = nullptr;
        LinkQueue<BTreeNode<T>*> queue;

        switch (order)
        {
        case PreOrder:
            PreOrderTraversal(root(), queue);
            break;
        case InOrder:
            InOrderTraversal(root(), queue);
            break;
        case PostOrder:
            PostOrderTraversal(root(), queue);
            break;
        }

        ret = new DynamicArray<T>(queue.length());

        if (ret != nullptr)
        {
            for (int i=0; i<ret->length(); ++i, queue.remove())
            {
                ret->set(i, queue.front()->value);
            }
        }
        else
        {
            THROW_EXCEPTION(NoEnoughMemoryException, "No enough to create return array ...");
        }

        return ret;
    }

    ~BTree()
    {
        clear();
    }

protected:
    LinkQueue<BTreeNode<T>*> m_queue;

    BTree(const BTree<T>&) = default;
    BTree<T>& operator = (const BTree<T>&) = default;

    virtual BTreeNode<T>* find(BTreeNode<T> *node, const T &value) const
    {
        BTreeNode<T> *ret = nullptr;

        if (node != nullptr)
        {
            if (node->value == value)
            {
                ret = node;
            }
            else
            {
                if (ret == nullptr)
                {
                    ret = find(node->left, value);
                }

                if (ret == nullptr)
                {
                    ret = find(node->right, value);
                }
            }
        }

        return ret;
    }

    virtual BTreeNode<T>* find(BTreeNode<T> *node, BTreeNode<T> *obj) const
    {
        BTreeNode<T> *ret = nullptr;

        if (node == obj)
        {
            ret = node;
        }
        else
        {
            if (node != nullptr)
            {
                if (ret == nullptr)
                {
                    ret = find(node->left, obj);
                }

                if (ret == nullptr)
                {
                    ret = find(node->right, obj);
                }
            }
        }

        return ret;
    }

    virtual bool insert(BTreeNode<T> *node, BTreeNode<T> *np, BTNodePos pos)
    {
        bool ret = true;

        if (pos == ANY)
        {
            if (np->left == nullptr)
            {
                np->left = node;
            }
            else if (np->right == nullptr)
            {
                np->right = node;
            }
            else
            {
                ret = false;
            }
        }
        else if (pos == LEFT)
        {
            if (np->left == nullptr)
            {
                np->left = node;
            }
            else
            {
                ret = false;
            }
        }
        else if (pos == RIGHT)
        {
            if (np->right == nullptr)
            {
                np->right = node;
            }
            else
            {
                ret = false;
            }
        }

        return ret;
    }

    virtual void remove(BTreeNode<T> *node, BTree<T> *&ret)
    {
        ret = new BTree<T>();

        if (ret != nullptr)
        {
            if (root() == node)
            {
                this->m_root = nullptr;
            }
            else
            {
                BTreeNode<T> *parent = dynamic_cast<BTreeNode<T>*>(node->parent);

                if (node == parent->left)
                {
                    parent->left = nullptr;
                }
                else if (node == parent->right)
                {
                    parent->right = nullptr;
                }

                node->parent = nullptr;
            }

            ret->m_root = node;
        }
        else
        {
            THROW_EXCEPTION(NoEnoughMemoryException, "No memory to create btree ...");
        }
    }

    virtual void free(BTreeNode<T> *node)
    {
        if (node != nullptr)
        {
            free(node->left);
            free(node->right);

            if (node->flag())
            {
                delete node;
            }
        }
    }

    int count(BTreeNode<T> *node) const
    {
        return (node != nullptr) ? (count(node->left) + count(node->right) + 1) : 0;
    }

    int height(BTreeNode<T> *node) const
    {
        int ret = 0;

        if (node != nullptr)
        {
            int lh = height(node->left);
            int rh = height(node->right);

            ret = ((lh > rh) ? lh : rh) + 1;
        }

        return ret;
    }

    int degree(BTreeNode<T> *node) const
    {
        int ret = 0;

        if (node != nullptr)
        {
            BTreeNode<T> *child[] = {node->left, node->right};
            ret = !!node->left + !!node->left;

            for (int i=0; (i<2) && (ret<2); ++i)
            {
                int d = degree(child[i]);

                if (ret < d)
                {
                    ret = d;
                }
            }
        }

        return ret;
    }

    void PreOrderTraversal(BTreeNode<T> *node, LinkQueue<BTreeNode<T>*> &queue) const
    {
        if (node != nullptr)
        {
            queue.add(node);
            PreOrderTraversal(node->left, queue);
            PreOrderTraversal(node->right, queue);
        }
    }

    void InOrderTraversal(BTreeNode<T> *node, LinkQueue<BTreeNode<T>*> &queue) const
    {
        if (node != nullptr)
        {
            InOrderTraversal(node->left, queue);
            queue.add(node);
            InOrderTraversal(node->right, queue);
        }
    }

    void PostOrderTraversal(BTreeNode<T> *node, LinkQueue<BTreeNode<T>*> &queue) const
    {
        if (node != nullptr)
        {
            PostOrderTraversal(node->left, queue);
            PostOrderTraversal(node->right, queue);
            queue.add(node);
        }
    }
};

}

#endif // BTREE_H

文件：main.cpp

#include <iostream>
#include "BTreeNode.h"
#include "BTree.h"

using namespace std;
using namespace DTLib;

int main()
{
    BTree<int> bt;
    BTreeNode<int> *n = nullptr;

    bt.insert(1, nullptr);

    n = bt.find(1);
    bt.insert(2, n);
    bt.insert(3, n);

    n = bt.find(2);
    bt.insert(4, n);
    bt.insert(5, n);

    n = bt.find(4);
    bt.insert(8, n);
    bt.insert(9, n);

    n = bt.find(5);
    bt.insert(10, n);

    n = bt.find(3);
    bt.insert(6, n);
    bt.insert(7, n);

    for (bt.begin(); !bt.end(); bt.next())
    {
        cout << bt.current() << " ";
    }

    cout << endl;

    SharedPointer<DynamicArray<int>> sp1 = bt.traversal(PreOrder);
    for (int i=0; i<(*sp1).length(); ++i)
    {
        cout << (*sp1)[i] << " ";
    }

    cout << endl;

    SharedPointer<DynamicArray<int>> sp2 = bt.traversal(InOrder);
    for (int i=0; i<(*sp2).length(); ++i)
    {
        cout << (*sp2)[i] << " ";
    }

    cout << endl;

    SharedPointer<DynamicArray<int>> sp3 = bt.traversal(PostOrder);
    for (int i=0; i<(*sp3).length(); ++i)
    {
        cout << (*sp3)[i] << " ";
    }

    cout << endl;

    return 0;
}

输出：

1 2 3 4 5 6 7 8 9 10
1 2 4 8 9 5 10 3 6 7
8 4 9 2 10 5 1 6 3 7
8 9 4 10 5 2 6 7 3 1

小结

• 二叉树的典型遍历都是以递归方式执行的
• BTree 以不同的函数接口支持典型遍历
• 层次遍历与典型遍历互不冲突
• 遍历结果能够反映树结点的先后次序

以上内容整理于狄泰软件学院系列课程，请大家保护原创！