# 2.4 - Binary Search Tree

### 1. 定義

* 是一個binary tree，可以為空
* left-children < root
* right-children > root
* left-children和right-children都是binary search tree

```
     10
    /  \
   5   12
  / \    \
 3   8    15
         /
        14
```

### 2. 相關演算法

#### 2.1 在二元搜尋樹中尋找x

1. 若b是空樹，搜尋失敗。
2. 若x等於b的根節點的資料域之值，尋找成功。
3. 若x小於b的根節點的資料域之值，搜尋左子樹。
4. 若x大於b的根節點的資料域之值，尋找右子樹。

```
bool SearchBST(bt_tree *bst, int x){
    if (bst != NULL){
        if (x > bst->Data) SearchBST(bt->rchild, x);
        if (x < bst->Data) SearchBST(bt->lchild, x);
        if (x == bst->Data) return True;
    }
    return False;
}
```

時間複雜度分析：

1. worst case: O(n) in skewed binary tree
2. best case: O(logn) in complete/full binary tree

#### 2.2 在二元搜尋樹中刪除節點x

1. 在BST中尋找x的位置
2. 如果x是leaf，直接刪除
3. 如果x是degree為1的節點
   1. 刪除x
   2. 將x的父點指向x的pointer指向x的子點
4. 如果x是degree為2的節點
   1. 找到x的左子樹中的最大值/右子樹中的最小值y
   2. y回到step2/step3

```
void DeleteBST(bt_tree *bst, int x){
    if (bst != NULL){
        if (x > bst->Data) SearchBST(bt->rchild, x);
        if (x < bst->Data) SearchBST(bt->lchild, x);
        if (x == bst->Data) return Delete(bst);
    }
}

void Delete(bt_tree *bst){
    bt_tree *temp;
    if (bst->lchild == NULL && bst->child == NULL){
        bst = NULL;
    }
    else if (bst->lchild != NULL && bst->rchild == NULL){
        temp = bst->lchild;
        bst->Data = temp->Data;
        bst->lchild = temp->lchild;
        bst->rchild = temp->rchild;
        temp = NULL;
    }
    else if (bst->lchild == NULL && bst->rchild != NULL){
        temp = bst->rchild;
        bst->Data = temp->Data; //將子節點的值取代x
        bst->lchild = temp->lchild;
        bst->rchild = temp->rchild;
        temp = NULL;
    }
    else{ //以左子樹的最大值取代x
        bt_tree *parent = bst;
        temp = bst->lchild;
        while(temp->rchild != NULL){
            parent = temp;
            temp = temp->rchild; //左子樹的最大值
        }
        bt->Data = temp->Data;
        if (bst != parent){ 
            parent->rchild = temp->lchild;
        }
        else{  //左子樹是skewed binary tree
            parent->lchild = temp->lchild;
        }
    }
}
```

#### 2.3 在二元搜尋樹中加入節點x

1. 若是空樹，則將x作為root插入
2. 若x小於根節點，則把x插入到左子樹中
3. 若x大於根節點，則把x插入到右子樹中（新插入節點總是在leaf）

```
void insertBST(bt_tree *bst, int x){
    if (bst == NULL){
        bst->Data = x;
    }
    else{
        if (x > bst->Data){
            insertBST(bst->rchild, x);
        }
        else{
            insertBST(bst->lchild, x);
        }
    }
}
```


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