二叉排序樹BST及CRUD操作

摘要#

構造一顆二叉排序樹(也叫二叉搜尋樹，BST，Binary Search Tree)十分簡單。一般來講，大於根節點的放在根節點的右子樹上，小於根節點的放在根節點的左子樹上(如果等於根節點，則可視情況而定)，如果寫程式的話，可以採用遞迴的方式，而且由於不存在重疊子問題的情況，因此遞迴的效能已經足夠好(不考慮棧溢位的情況)。

二叉排序樹在通常情況下可以達到O(lgN)的靜態、動態操作的時間複雜度，但是存在一種特殊情況，即輸入的資料本身就是有序的，這時二叉排序樹退化成向量。

下面我們系統歸納一下二叉樹的特性，以及相關操作及其程式碼實現。

二叉排序樹#

簡稱BST，也稱為二叉查詢樹。其或是一棵空樹，或是一棵具有下列特性的非空二叉樹：

1）若左子樹非空，則左子樹上所有結點關鍵字值均小於根結點的關鍵字值。

2）若右子樹非空，則右子樹上所有結點關鍵字值均大於根結點的關鍵字值。

3）左、右子樹本身也分別是一棵二叉排序樹。

其是一個遞迴的資料結構。

左子樹結點值 < 根結點值 < 右子樹結點值

對其進行中序遍歷可以得到一個遞增的有序序列。

圖1. 二叉排序樹示例圖

CRUD操作#

Create-構造二叉排序樹#

構造一棵二叉排序樹就是依次輸入資料元素，並將它們插入到二叉樹中的適當位置上的過程。

具體過程：

1）每讀入一個元素，就建立一個新節點。

2）若二叉排序樹非空，則將新結點的值與根結點的值比較。如果小於根結點的值，則插入到左子樹中，否則插入到右子樹中。

3）若二叉排序樹為空，則新結點作為二叉樹的根結點。

Copy

void Create_BST(BiTree &T, KeyType str[], int n) { //用關鍵字陣列str[]建立一個二叉排序樹 T = NULL; //初始時bt為空樹 int i = 0; while(i < n) { //依次將每個元素插入 BST_Insert(T, str[i]); i++; } }

Retrieve-查詢二叉排序樹的某結點#

二叉排序樹的查詢是從根結點開始，沿某一分支逐層向下進行比較的一個遞迴的過程。

具體查詢過程是：

1）若二叉樹非空，將給定值與根結點的關鍵字比較，若相等，則查詢成功；

2）若不等，則當根結點的關鍵字大於給定關鍵字值k時，在根結點的左子樹中查詢；

3）當根結點的關鍵字小於給定關鍵字值k時，在根結點的右子樹中查詢。

二叉排序樹的非遞迴查詢演算法：

Copy

BSTNode *BST_Search(BiTree T,ElemTypr key,BSTNode *&p) { //查詢函式返回值指向關鍵字值為key的結點指標，若不存在，返回NULL p = NULL; //p指向被查詢結點的雙親，用於插入和刪除操作中 while(T != NULL && key != T->data) { p = T; if(key < T->data) { T = T->lchild; } else { T = T->rchild; } return T; } }

Update-插入結點到二叉排序樹中#

二叉排序樹作為一種動態集合，其特點是樹的結構通常不是一次生成的，而是在查詢過程中，當樹中不存在關鍵字等於給定值的結點時再進行插入。

由於二叉排序樹是遞迴定義的，其插入結點的過程是：

1）若原二叉樹為空，則直接插入結點；

2）否則，若關鍵字k小於根結點關鍵字，則插入到左子樹中；

3）若關鍵字k大於根結點關鍵字，則插入到右子樹中。

Copy

int BST_Insert(BiTree &T, KeyType k) { //在二叉=排序樹T中插入一個關鍵字為k的結點 if(T == NULL) { T = (BiTree)malloc(sizeof(BSTNode)); T->key = k; T->lchild = T->rchild = NULL; return 1; //返回1，表示成功 } else if(k == T-> key) { //樹中存在相同關鍵字的結點 return 0; } else if(k < T->key) { //插入到T的左子樹中 return BST_Insert(T->lchild, k); } else { return BST_Insert(T->rchild, k); } }

由此可見，插入的新結點一定是某個葉結點。下圖是向二叉樹插入結點28的過程，其中虛線表示查詢路徑。

圖2. 向二叉排序樹插入結點28

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Delete-刪除二叉樹的結點#

在二叉排序樹中刪除一個結點時，不能把以該結點為根的子樹上的結點都刪除，必須把被刪除結點從儲存二叉排序樹的連結串列上摘下，將因刪除結點而斷開的二叉連結串列重新連結起來，同時確保二叉排序樹的性質不會丟失。

刪除操作的實現過程按3種情況來處理：

1）如果被刪除結點z是葉結點，則直接刪除，不會破壞二叉排序樹的性質。

2）若結點z只有一棵左子樹或右子樹，則讓z的子樹成為z父結點的子樹，替代z的位置。

3）若結點z有左、右兩棵子樹，則令z的直接後繼【中序第一個子女】（或直接前驅）替代z，然後從二叉排序樹中刪去這個直接後繼【中序第一個子女】（或直接前驅），這樣就轉換成了第一或第二種情況。

圖3. 3種情況下的刪除過程

參考#

[1] miao_zheng. 二叉排序樹、平衡二叉樹、B樹&B+樹、紅黑樹的設計動機、缺陷與應用場景[OL]. cnblogs, 2018-02-28/2020-06-20
[2] 王道論壇. 2019年資料結構考研複習指導[M].北京:電子工業出版社, 2018:153-155.