二叉搜索樹BST

在二叉搜索樹b中查找x的過程爲：

1.若b是空樹，則搜索失敗，否則：

2.若x等於b的根結點的數據域之值，則查找成功；否則

3.若x小於b的根結點的數據域之值，則搜索左子樹：否則

4.查找右子樹

// 指針parent指向pRoot的父節點，其初始調用值爲NULL
// 若查找成功，指針pTarget指向目標節點，函數返回true
// 否則指針pTarget指向查找路徑上訪問的最後一個節點，函數返回false
// pTarget初始值爲NULL，由於函數調用過程中需要修改其指針值，所以參數傳遞雙層指針
bool searchBST(BinaryTreeNode* pRoot, int key, BinaryTreeNode* parent, BinaryTreeNode** pTarget)
{
	if(pRoot == NULL)
	{
		*pTarget = parent;
		return false;
	}

	if(pRoot->value == key)
	{
		*pTarget = pRoot;
		return true;
	}
	else if(pRoot->value > key)
		return searchBST(pRoot->left, key, pRoot, pTarget);
	else
		return searchBST(pRoot->right, key, pRoot, pTarget);
}

向一個二叉搜索樹b中插入一個節點s的算法，過程爲：

1.若b是空樹，則將s所指節點作爲根節點插入，否則：

2.若s->value等於b的根節點的數據域之值，則返回，否則：

3.若s->value小於b的根節點的數據域之值，則把s所指節點插入到左子樹中，否則：

4.把s所指節點插入到右子樹中。（新插入節點總是葉子節點）

// 當樹爲空時，需要修改根結點指針值，所以參數傳遞雙層指針
bool insertBST(BinaryTreeNode** pRoot, int key)
{
	if(*pRoot == NULL)
	{
		BinaryTreeNode* s = new BinaryTreeNode;
		s->value = key;
		s->left = s->right = NULL;
		*pRoot = s;
		return true;
	}
	else if((*pRoot)->value == key)
		return false;

	if((*pRoot)->value > key)
		return insertBST(&(*pRoot)->left, key);
	else
		return insertBST(&(*pRoot)->right, key);
}

在二叉查找樹中刪去一個節點，分三種情況討論：

1.若*p節點爲葉子節點，即PL（左子樹）和PR（右子樹）均爲空樹。由於刪去葉子節點不破壞整棵樹的結構，則只需修改其雙親節點的指針即可。

2.若*p節點只有左子樹PL或右子樹PR，此時只要另PL或PR直接成爲其雙親節點*f的左子樹（當*p是左子樹）或右子樹（當*p是右子樹）即可。

3.若*p節點的左子樹和右子樹均不空。在刪去*p之後，爲保持其它元素之間的相對位置不變，可按中序遍歷保持有序進行調整，可以有兩種做法：其一是另*p的直接前驅（inorder predecessor）或直接後繼（inorder successor）替代*p，然後再從二叉查找樹中刪去它的直接前驅（或直接後繼）

bool deleteNode(BinaryTreeNode** pRoot)
{
	BinaryTreeNode* q, *s;
	if((*pRoot)->left == NULL && (*pRoot)->right == NULL)
	{
		delete *pRoot;
		*pRoot = NULL;
	}
	else if((*pRoot)->right == NULL)
	{
		q = (*pRoot)->left;
		(*pRoot)->value = q->value;
		(*pRoot)->left = q->left;
		(*pRoot)->right = q->right;

		delete q;
	}
	else if((*pRoot)->left == NULL)
	{
		q = (*pRoot)->right;
		(*pRoot)->value = q->value;
		(*pRoot)->left = q->left;
		(*pRoot)->right = q->right;

		delete q;
	}
	else
	{
		q = *pRoot;
		s = (*pRoot)->left;
		while(s->right)
		{
			q = s;
			s = s->right;
		}
		(*pRoot)->value = s->value;
		if(q != *pRoot)
			q->right = s->left;
		else
			q->left = s->left;

		delete s;
	}

	return true;
}

bool deleteBST(BinaryTreeNode** pRoot, int key)
{
	if(*pRoot == NULL)
		return false;

	if((*pRoot)->value == key)
		return deleteNode(pRoot);
	else if((*pRoot)->value > key)
		return deleteBST(&(*pRoot)->left, key);
	else
		return deleteBST(&(*pRoot)->right, key);
}

完整驗證代碼：

#include <iostream>

using namespace std;

struct BinaryTreeNode
{
	int value;
	BinaryTreeNode* left, *right;
};

// 指針parent指向pRoot的父節點，其初始調用值爲NULL
// 若查找成功，指針pTarget指向目標節點，函數返回true
// 否則指針pTarget指向查找路徑上訪問的最後一個節點，函數返回false
// pTarget初始值爲NULL，由於函數調用過程中需要修改其指針值，所以參數傳遞雙層指針
bool searchBST(BinaryTreeNode* pRoot, int key, BinaryTreeNode* parent, BinaryTreeNode** pTarget)
{
	if(pRoot == NULL)
	{
		*pTarget = parent; 
		return false;
	}

	if(pRoot->value == key)
	{
		*pTarget = pRoot;
		return true;
	}
	else if(pRoot->value > key)
		return searchBST(pRoot->left, key, pRoot, pTarget);
	else
		return searchBST(pRoot->right, key, pRoot, pTarget);
}

// 當樹爲空時，需要修改根結點指針值，所以參數傳遞雙層指針
bool insertBST(BinaryTreeNode** pRoot, int key)
{
	if(*pRoot == NULL)
	{
		BinaryTreeNode* s = new BinaryTreeNode;
		s->value = key;
		s->left = s->right = NULL;
		*pRoot = s;
		return true;
	}
	else if((*pRoot)->value == key)
		return false;

	if((*pRoot)->value > key)
		return insertBST(&(*pRoot)->left, key);
	else
		return insertBST(&(*pRoot)->right, key);
}

bool deleteNode(BinaryTreeNode** pRoot)
{
	BinaryTreeNode* q, *s;
	if((*pRoot)->left == NULL && (*pRoot)->right == NULL)
	{
		delete *pRoot;
		*pRoot = NULL;
	}
	else if((*pRoot)->right == NULL)
	{
		q = (*pRoot)->left;
		(*pRoot)->value = q->value;
		(*pRoot)->left = q->left;
		(*pRoot)->right = q->right;

		delete q;
	}
	else if((*pRoot)->left == NULL)
	{
		q = (*pRoot)->right;
		(*pRoot)->value = q->value;
		(*pRoot)->left = q->left;
		(*pRoot)->right = q->right;

		delete q;
	}
	else
	{
		q = *pRoot;
		s = (*pRoot)->left;
		while(s->right)
		{
			q = s;
			s = s->right;
		}
		(*pRoot)->value = s->value;
		if(q != *pRoot)
			q->right = s->left;
		else
			q->left = s->left;

		delete s;
	}

	return true;
}

bool deleteBST(BinaryTreeNode** pRoot, int key)
{
	if(*pRoot == NULL)
		return false;

	if((*pRoot)->value == key)
		return deleteNode(pRoot);
	else if((*pRoot)->value > key)
		return deleteBST(&(*pRoot)->left, key);
	else
		return deleteBST(&(*pRoot)->right, key);
}

void printInorder(BinaryTreeNode* pRoot)
{
	if(pRoot->left != NULL)
		printInorder(pRoot->left);

	printf("%d ", pRoot->value);

	if(pRoot->right != NULL)
		printInorder(pRoot->right);
}

int main()
{
	int a[10] = {62, 88, 58, 47, 35, 73, 51, 99, 37, 93};
	BinaryTreeNode* pRoot = NULL;
	printf("Build the binary search tree:\n");
	for(int i = 0; i < 10; i++)
	{
		insertBST(&pRoot, a[i]);
		printInorder(pRoot);
		printf("\n");
	}

    printf("which node do you want to delete?\n");
	int num = 0;
	while(scanf("%d", &num))
	{

		deleteBST(&pRoot, num);
		printInorder(pRoot);
		printf("\n");
	}

	return 0;
}