2023-02-28 20:03:53 +08:00
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# 二叉树
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2022-11-22 17:47:26 +08:00
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2023-08-20 23:28:30 +08:00
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「二叉树 binary tree」是一种非线性数据结构,代表着祖先与后代之间的派生关系,体现着“一分为二”的分治逻辑。与链表类似,二叉树的基本单元是节点,每个节点包含:值、左子节点引用、右子节点引用。
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2022-11-22 17:47:26 +08:00
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2023-09-04 03:19:08 +08:00
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=== "Python"
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2022-11-22 17:47:26 +08:00
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2023-09-04 03:19:08 +08:00
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```python title=""
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class TreeNode:
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"""二叉树节点类"""
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def __init__(self, val: int):
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2023-09-20 01:53:44 +08:00
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self.val: int = val # 节点值
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2023-09-19 14:41:02 +08:00
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self.left: TreeNode | None = None # 左子节点引用
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self.right: TreeNode | None = None # 右子节点引用
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2022-11-22 17:47:26 +08:00
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```
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2022-11-29 02:21:49 +08:00
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=== "C++"
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2022-12-03 01:31:29 +08:00
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```cpp title=""
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2023-04-09 04:32:17 +08:00
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/* 二叉树节点结构体 */
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2022-11-29 02:21:49 +08:00
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struct TreeNode {
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2023-04-09 04:32:17 +08:00
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int val; // 节点值
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TreeNode *left; // 左子节点指针
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TreeNode *right; // 右子节点指针
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2022-11-29 02:21:49 +08:00
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TreeNode(int x) : val(x), left(nullptr), right(nullptr) {}
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};
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```
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2023-09-04 03:19:08 +08:00
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=== "Java"
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2022-11-29 02:21:49 +08:00
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2023-09-04 03:19:08 +08:00
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```java title=""
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/* 二叉树节点类 */
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class TreeNode {
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int val; // 节点值
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TreeNode left; // 左子节点引用
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TreeNode right; // 右子节点引用
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TreeNode(int x) { val = x; }
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}
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```
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=== "C#"
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```csharp title=""
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/* 二叉树节点类 */
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class TreeNode {
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int val; // 节点值
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TreeNode? left; // 左子节点引用
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TreeNode? right; // 右子节点引用
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TreeNode(int x) { val = x; }
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}
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2022-11-29 02:21:49 +08:00
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```
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=== "Go"
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2022-12-03 01:31:29 +08:00
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```go title=""
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2023-04-09 04:32:17 +08:00
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/* 二叉树节点结构体 */
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2022-12-03 20:25:24 +08:00
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type TreeNode struct {
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Val int
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Left *TreeNode
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Right *TreeNode
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}
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2023-04-09 04:32:17 +08:00
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/* 节点初始化方法 */
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2022-12-03 20:25:24 +08:00
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func NewTreeNode(v int) *TreeNode {
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return &TreeNode{
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2023-08-17 05:13:19 +08:00
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Left: nil, // 左子节点指针
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Right: nil, // 右子节点指针
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Val: v, // 节点值
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2022-12-03 20:25:24 +08:00
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}
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}
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2022-12-03 01:31:29 +08:00
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```
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2023-09-04 03:19:08 +08:00
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=== "Swift"
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```swift title=""
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/* 二叉树节点类 */
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class TreeNode {
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var val: Int // 节点值
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var left: TreeNode? // 左子节点引用
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var right: TreeNode? // 右子节点引用
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init(x: Int) {
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val = x
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}
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}
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```
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2023-07-26 15:35:38 +08:00
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=== "JS"
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2022-12-03 01:31:29 +08:00
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2023-02-08 04:27:55 +08:00
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```javascript title=""
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2023-04-09 04:32:17 +08:00
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/* 二叉树节点类 */
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2023-10-01 19:33:53 +08:00
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class TreeNode {
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val; // 节点值
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left; // 左子节点指针
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right; // 右子节点指针
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constructor(val, left, right) {
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this.val = val === undefined ? 0 : val;
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this.left = left === undefined ? null : left;
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this.right = right === undefined ? null : right;
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}
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2022-12-05 00:13:48 +08:00
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}
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2022-12-03 01:31:29 +08:00
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```
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2023-07-26 15:35:38 +08:00
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=== "TS"
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2022-12-03 01:31:29 +08:00
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```typescript title=""
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2023-04-09 04:32:17 +08:00
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/* 二叉树节点类 */
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2022-12-15 00:00:40 +08:00
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class TreeNode {
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val: number;
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left: TreeNode | null;
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right: TreeNode | null;
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2022-12-21 17:19:39 +08:00
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2022-12-15 00:00:40 +08:00
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constructor(val?: number, left?: TreeNode | null, right?: TreeNode | null) {
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2023-04-09 04:32:17 +08:00
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this.val = val === undefined ? 0 : val; // 节点值
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2023-08-17 05:13:19 +08:00
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this.left = left === undefined ? null : left; // 左子节点引用
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this.right = right === undefined ? null : right; // 右子节点引用
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2022-12-15 00:00:40 +08:00
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}
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}
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2022-12-03 01:31:29 +08:00
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```
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2023-09-04 03:19:08 +08:00
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=== "Dart"
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```dart title=""
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/* 二叉树节点类 */
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class TreeNode {
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int val; // 节点值
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TreeNode? left; // 左子节点引用
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TreeNode? right; // 右子节点引用
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TreeNode(this.val, [this.left, this.right]);
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}
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```
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=== "Rust"
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```rust title=""
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2023-10-01 19:25:03 +08:00
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use std::rc::Rc;
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use std::cell::RefCell;
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2023-09-04 03:19:08 +08:00
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2023-10-01 19:25:03 +08:00
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/* 二叉树节点类型 */
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struct TreeNode {
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val: i32, // 节点值
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left: Option<Rc<RefCell<TreeNode>>>, // 左子节点引用
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right: Option<Rc<RefCell<TreeNode>>>, // 右子节点引用
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}
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impl TreeNode {
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/* 二叉树节点构造方法 */
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fn new(val: i32) -> Rc<RefCell<Self>> {
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Rc::new(RefCell::new(Self {
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val,
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left: None,
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right: None
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}))
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}
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}
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2023-09-04 03:19:08 +08:00
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```
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2022-12-03 01:31:29 +08:00
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=== "C"
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```c title=""
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2023-04-18 20:21:31 +08:00
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/* 二叉树节点结构体 */
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struct TreeNode {
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int val; // 节点值
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int height; // 节点高度
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struct TreeNode *left; // 左子节点指针
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struct TreeNode *right; // 右子节点指针
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};
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typedef struct TreeNode TreeNode;
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/* 构造函数 */
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TreeNode *newTreeNode(int val) {
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TreeNode *node;
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node = (TreeNode *)malloc(sizeof(TreeNode));
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node->val = val;
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node->height = 0;
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node->left = NULL;
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node->right = NULL;
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return node;
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}
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2022-12-03 01:31:29 +08:00
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```
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2023-02-01 22:03:04 +08:00
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=== "Zig"
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```zig title=""
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```
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2023-08-20 23:28:30 +08:00
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每个节点都有两个引用(指针),分别指向「左子节点 left-child node」和「右子节点 right-child node」,该节点被称为这两个子节点的「父节点 parent node」。当给定一个二叉树的节点时,我们将该节点的左子节点及其以下节点形成的树称为该节点的「左子树 left subtree」,同理可得「右子树 right subtree」。
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2022-11-22 17:47:26 +08:00
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2023-08-21 19:33:45 +08:00
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**在二叉树中,除叶节点外,其他所有节点都包含子节点和非空子树**。如下图所示,如果将“节点 2”视为父节点,则其左子节点和右子节点分别是“节点 4”和“节点 5”,左子树是“节点 4 及其以下节点形成的树”,右子树是“节点 5 及其以下节点形成的树”。
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2022-12-25 14:05:42 +08:00
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2023-04-09 04:32:17 +08:00
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![父节点、子节点、子树](binary_tree.assets/binary_tree_definition.png)
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2022-11-22 17:47:26 +08:00
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2023-02-16 03:39:01 +08:00
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## 二叉树常见术语
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2022-11-22 17:47:26 +08:00
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2023-08-21 19:33:45 +08:00
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二叉树的常用术语如下图所示。
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2022-11-22 17:47:26 +08:00
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2023-08-20 23:28:30 +08:00
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- 「根节点 root node」:位于二叉树顶层的节点,没有父节点。
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- 「叶节点 leaf node」:没有子节点的节点,其两个指针均指向 $\text{None}$ 。
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- 「边 edge」:连接两个节点的线段,即节点引用(指针)。
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- 节点所在的「层 level」:从顶至底递增,根节点所在层为 1 。
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2023-08-27 23:40:39 +08:00
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- 节点的「度 degree」:节点的子节点的数量。在二叉树中,度的取值范围是 0、1、2 。
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2023-08-20 23:28:30 +08:00
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- 二叉树的「高度 height」:从根节点到最远叶节点所经过的边的数量。
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2023-08-27 23:40:39 +08:00
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- 节点的「深度 depth」:从根节点到该节点所经过的边的数量。
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2023-08-20 23:28:30 +08:00
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- 节点的「高度 height」:从最远叶节点到该节点所经过的边的数量。
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2022-11-22 17:47:26 +08:00
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2023-02-26 18:18:34 +08:00
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![二叉树的常用术语](binary_tree.assets/binary_tree_terminology.png)
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2022-11-22 17:47:26 +08:00
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2023-08-29 20:57:37 +08:00
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!!! tip
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2022-12-18 15:59:34 +08:00
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2023-08-20 23:28:30 +08:00
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请注意,我们通常将“高度”和“深度”定义为“走过边的数量”,但有些题目或教材可能会将其定义为“走过节点的数量”。在这种情况下,高度和深度都需要加 1 。
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2022-12-18 15:59:34 +08:00
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2023-02-16 03:39:01 +08:00
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## 二叉树基本操作
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2022-11-22 17:47:26 +08:00
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2023-08-21 19:33:45 +08:00
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### 初始化二叉树
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与链表类似,首先初始化节点,然后构建引用(指针)。
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2022-11-22 17:47:26 +08:00
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2023-09-04 03:19:08 +08:00
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=== "Python"
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2022-11-22 17:47:26 +08:00
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2023-09-04 03:19:08 +08:00
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```python title="binary_tree.py"
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# 初始化二叉树
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# 初始化节点
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n1 = TreeNode(val=1)
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n2 = TreeNode(val=2)
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n3 = TreeNode(val=3)
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n4 = TreeNode(val=4)
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n5 = TreeNode(val=5)
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# 构建引用指向(即指针)
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n1.left = n2
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n1.right = n3
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n2.left = n4
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n2.right = n5
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2022-11-22 17:47:26 +08:00
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```
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2022-11-29 02:21:49 +08:00
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=== "C++"
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```cpp title="binary_tree.cpp"
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/* 初始化二叉树 */
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2023-04-09 04:32:17 +08:00
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// 初始化节点
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2022-11-29 02:21:49 +08:00
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TreeNode* n1 = new TreeNode(1);
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TreeNode* n2 = new TreeNode(2);
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TreeNode* n3 = new TreeNode(3);
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TreeNode* n4 = new TreeNode(4);
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TreeNode* n5 = new TreeNode(5);
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// 构建引用指向(即指针)
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n1->left = n2;
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n1->right = n3;
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n2->left = n4;
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n2->right = n5;
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```
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2023-09-04 03:19:08 +08:00
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=== "Java"
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2022-11-29 02:21:49 +08:00
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|
2023-09-04 03:19:08 +08:00
|
|
|
|
```java title="binary_tree.java"
|
|
|
|
|
// 初始化节点
|
|
|
|
|
TreeNode n1 = new TreeNode(1);
|
|
|
|
|
TreeNode n2 = new TreeNode(2);
|
|
|
|
|
TreeNode n3 = new TreeNode(3);
|
|
|
|
|
TreeNode n4 = new TreeNode(4);
|
|
|
|
|
TreeNode n5 = new TreeNode(5);
|
|
|
|
|
// 构建引用指向(即指针)
|
|
|
|
|
n1.left = n2;
|
|
|
|
|
n1.right = n3;
|
|
|
|
|
n2.left = n4;
|
|
|
|
|
n2.right = n5;
|
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
=== "C#"
|
|
|
|
|
|
|
|
|
|
```csharp title="binary_tree.cs"
|
|
|
|
|
/* 初始化二叉树 */
|
|
|
|
|
// 初始化节点
|
|
|
|
|
TreeNode n1 = new TreeNode(1);
|
|
|
|
|
TreeNode n2 = new TreeNode(2);
|
|
|
|
|
TreeNode n3 = new TreeNode(3);
|
|
|
|
|
TreeNode n4 = new TreeNode(4);
|
|
|
|
|
TreeNode n5 = new TreeNode(5);
|
|
|
|
|
// 构建引用指向(即指针)
|
|
|
|
|
n1.left = n2;
|
|
|
|
|
n1.right = n3;
|
|
|
|
|
n2.left = n4;
|
|
|
|
|
n2.right = n5;
|
2022-11-29 02:21:49 +08:00
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
=== "Go"
|
|
|
|
|
|
|
|
|
|
```go title="binary_tree.go"
|
2022-12-03 20:25:24 +08:00
|
|
|
|
/* 初始化二叉树 */
|
2023-04-09 04:32:17 +08:00
|
|
|
|
// 初始化节点
|
2022-12-03 20:25:24 +08:00
|
|
|
|
n1 := NewTreeNode(1)
|
|
|
|
|
n2 := NewTreeNode(2)
|
|
|
|
|
n3 := NewTreeNode(3)
|
|
|
|
|
n4 := NewTreeNode(4)
|
|
|
|
|
n5 := NewTreeNode(5)
|
|
|
|
|
// 构建引用指向(即指针)
|
|
|
|
|
n1.Left = n2
|
|
|
|
|
n1.Right = n3
|
|
|
|
|
n2.Left = n4
|
|
|
|
|
n2.Right = n5
|
2022-11-29 02:21:49 +08:00
|
|
|
|
```
|
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
=== "Swift"
|
|
|
|
|
|
|
|
|
|
```swift title="binary_tree.swift"
|
|
|
|
|
// 初始化节点
|
|
|
|
|
let n1 = TreeNode(x: 1)
|
|
|
|
|
let n2 = TreeNode(x: 2)
|
|
|
|
|
let n3 = TreeNode(x: 3)
|
|
|
|
|
let n4 = TreeNode(x: 4)
|
|
|
|
|
let n5 = TreeNode(x: 5)
|
|
|
|
|
// 构建引用指向(即指针)
|
|
|
|
|
n1.left = n2
|
|
|
|
|
n1.right = n3
|
|
|
|
|
n2.left = n4
|
|
|
|
|
n2.right = n5
|
|
|
|
|
```
|
|
|
|
|
|
2023-07-26 15:35:38 +08:00
|
|
|
|
=== "JS"
|
2022-12-03 01:31:29 +08:00
|
|
|
|
|
2023-02-08 04:27:55 +08:00
|
|
|
|
```javascript title="binary_tree.js"
|
2022-12-04 12:26:09 +08:00
|
|
|
|
/* 初始化二叉树 */
|
2023-04-09 04:32:17 +08:00
|
|
|
|
// 初始化节点
|
2022-12-04 12:26:09 +08:00
|
|
|
|
let n1 = new TreeNode(1),
|
2022-12-13 21:44:14 +08:00
|
|
|
|
n2 = new TreeNode(2),
|
|
|
|
|
n3 = new TreeNode(3),
|
|
|
|
|
n4 = new TreeNode(4),
|
|
|
|
|
n5 = new TreeNode(5);
|
2022-12-04 12:26:09 +08:00
|
|
|
|
// 构建引用指向(即指针)
|
|
|
|
|
n1.left = n2;
|
|
|
|
|
n1.right = n3;
|
|
|
|
|
n2.left = n4;
|
|
|
|
|
n2.right = n5;
|
2022-12-03 01:31:29 +08:00
|
|
|
|
```
|
|
|
|
|
|
2023-07-26 15:35:38 +08:00
|
|
|
|
=== "TS"
|
2022-12-03 01:31:29 +08:00
|
|
|
|
|
|
|
|
|
```typescript title="binary_tree.ts"
|
2022-12-15 00:00:40 +08:00
|
|
|
|
/* 初始化二叉树 */
|
2023-04-09 04:32:17 +08:00
|
|
|
|
// 初始化节点
|
2022-12-15 00:00:40 +08:00
|
|
|
|
let n1 = new TreeNode(1),
|
|
|
|
|
n2 = new TreeNode(2),
|
|
|
|
|
n3 = new TreeNode(3),
|
|
|
|
|
n4 = new TreeNode(4),
|
|
|
|
|
n5 = new TreeNode(5);
|
|
|
|
|
// 构建引用指向(即指针)
|
|
|
|
|
n1.left = n2;
|
|
|
|
|
n1.right = n3;
|
|
|
|
|
n2.left = n4;
|
|
|
|
|
n2.right = n5;
|
2022-12-03 01:31:29 +08:00
|
|
|
|
```
|
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
=== "Dart"
|
2022-12-03 01:31:29 +08:00
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
```dart title="binary_tree.dart"
|
2022-12-23 15:42:02 +08:00
|
|
|
|
/* 初始化二叉树 */
|
2023-04-09 04:32:17 +08:00
|
|
|
|
// 初始化节点
|
2022-12-23 15:42:02 +08:00
|
|
|
|
TreeNode n1 = new TreeNode(1);
|
|
|
|
|
TreeNode n2 = new TreeNode(2);
|
|
|
|
|
TreeNode n3 = new TreeNode(3);
|
|
|
|
|
TreeNode n4 = new TreeNode(4);
|
|
|
|
|
TreeNode n5 = new TreeNode(5);
|
|
|
|
|
// 构建引用指向(即指针)
|
|
|
|
|
n1.left = n2;
|
|
|
|
|
n1.right = n3;
|
|
|
|
|
n2.left = n4;
|
|
|
|
|
n2.right = n5;
|
2022-12-03 01:31:29 +08:00
|
|
|
|
```
|
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
=== "Rust"
|
2023-02-01 22:03:04 +08:00
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
```rust title="binary_tree.rs"
|
2023-10-01 19:25:03 +08:00
|
|
|
|
// 初始化节点
|
|
|
|
|
let n1 = TreeNode::new(1);
|
|
|
|
|
let n2 = TreeNode::new(2);
|
|
|
|
|
let n3 = TreeNode::new(3);
|
|
|
|
|
let n4 = TreeNode::new(4);
|
|
|
|
|
let n5 = TreeNode::new(5);
|
|
|
|
|
// 构建引用指向(即指针)
|
|
|
|
|
n1.borrow_mut().left = Some(n2.clone());
|
|
|
|
|
n1.borrow_mut().right = Some(n3);
|
|
|
|
|
n2.borrow_mut().left = Some(n4);
|
|
|
|
|
n2.borrow_mut().right = Some(n5);
|
2023-02-01 22:03:04 +08:00
|
|
|
|
```
|
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
=== "C"
|
2023-06-02 02:40:26 +08:00
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
```c title="binary_tree.c"
|
2023-06-02 14:56:29 +08:00
|
|
|
|
/* 初始化二叉树 */
|
|
|
|
|
// 初始化节点
|
2023-09-04 03:19:08 +08:00
|
|
|
|
TreeNode *n1 = newTreeNode(1);
|
|
|
|
|
TreeNode *n2 = newTreeNode(2);
|
|
|
|
|
TreeNode *n3 = newTreeNode(3);
|
|
|
|
|
TreeNode *n4 = newTreeNode(4);
|
|
|
|
|
TreeNode *n5 = newTreeNode(5);
|
2023-06-02 14:56:29 +08:00
|
|
|
|
// 构建引用指向(即指针)
|
2023-09-04 03:19:08 +08:00
|
|
|
|
n1->left = n2;
|
|
|
|
|
n1->right = n3;
|
|
|
|
|
n2->left = n4;
|
|
|
|
|
n2->right = n5;
|
2023-06-02 02:40:26 +08:00
|
|
|
|
```
|
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
=== "Zig"
|
2023-07-26 11:00:53 +08:00
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
```zig title="binary_tree.zig"
|
2023-07-26 11:00:53 +08:00
|
|
|
|
|
|
|
|
|
```
|
|
|
|
|
|
2023-08-22 13:51:25 +08:00
|
|
|
|
### 插入与删除节点
|
2023-08-21 19:33:45 +08:00
|
|
|
|
|
|
|
|
|
与链表类似,在二叉树中插入与删除节点可以通过修改指针来实现。下图给出了一个示例。
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2023-04-09 04:32:17 +08:00
|
|
|
|
![在二叉树中插入与删除节点](binary_tree.assets/binary_tree_add_remove.png)
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
=== "Python"
|
2022-11-29 02:21:49 +08:00
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
```python title="binary_tree.py"
|
|
|
|
|
# 插入与删除节点
|
|
|
|
|
p = TreeNode(0)
|
|
|
|
|
# 在 n1 -> n2 中间插入节点 P
|
|
|
|
|
n1.left = p
|
|
|
|
|
p.left = n2
|
|
|
|
|
# 删除节点 P
|
|
|
|
|
n1.left = n2
|
2022-11-29 02:21:49 +08:00
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
=== "C++"
|
|
|
|
|
|
|
|
|
|
```cpp title="binary_tree.cpp"
|
2023-04-09 04:32:17 +08:00
|
|
|
|
/* 插入与删除节点 */
|
2022-11-29 02:21:49 +08:00
|
|
|
|
TreeNode* P = new TreeNode(0);
|
2023-04-09 04:32:17 +08:00
|
|
|
|
// 在 n1 -> n2 中间插入节点 P
|
2022-11-29 02:21:49 +08:00
|
|
|
|
n1->left = P;
|
|
|
|
|
P->left = n2;
|
2023-04-09 04:32:17 +08:00
|
|
|
|
// 删除节点 P
|
2022-11-29 02:21:49 +08:00
|
|
|
|
n1->left = n2;
|
|
|
|
|
```
|
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
=== "Java"
|
2022-11-29 02:21:49 +08:00
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
```java title="binary_tree.java"
|
|
|
|
|
TreeNode P = new TreeNode(0);
|
|
|
|
|
// 在 n1 -> n2 中间插入节点 P
|
|
|
|
|
n1.left = P;
|
|
|
|
|
P.left = n2;
|
|
|
|
|
// 删除节点 P
|
|
|
|
|
n1.left = n2;
|
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
=== "C#"
|
|
|
|
|
|
|
|
|
|
```csharp title="binary_tree.cs"
|
|
|
|
|
/* 插入与删除节点 */
|
|
|
|
|
TreeNode P = new TreeNode(0);
|
|
|
|
|
// 在 n1 -> n2 中间插入节点 P
|
|
|
|
|
n1.left = P;
|
|
|
|
|
P.left = n2;
|
|
|
|
|
// 删除节点 P
|
|
|
|
|
n1.left = n2;
|
2022-11-29 02:21:49 +08:00
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
=== "Go"
|
|
|
|
|
|
|
|
|
|
```go title="binary_tree.go"
|
2023-04-09 04:32:17 +08:00
|
|
|
|
/* 插入与删除节点 */
|
|
|
|
|
// 在 n1 -> n2 中间插入节点 P
|
2022-12-03 20:25:24 +08:00
|
|
|
|
p := NewTreeNode(0)
|
|
|
|
|
n1.Left = p
|
|
|
|
|
p.Left = n2
|
2023-04-09 04:32:17 +08:00
|
|
|
|
// 删除节点 P
|
2022-12-03 20:25:24 +08:00
|
|
|
|
n1.Left = n2
|
2022-11-29 02:21:49 +08:00
|
|
|
|
```
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
=== "Swift"
|
|
|
|
|
|
|
|
|
|
```swift title="binary_tree.swift"
|
|
|
|
|
let P = TreeNode(x: 0)
|
|
|
|
|
// 在 n1 -> n2 中间插入节点 P
|
|
|
|
|
n1.left = P
|
|
|
|
|
P.left = n2
|
|
|
|
|
// 删除节点 P
|
|
|
|
|
n1.left = n2
|
|
|
|
|
```
|
|
|
|
|
|
2023-07-26 15:35:38 +08:00
|
|
|
|
=== "JS"
|
2022-12-03 01:31:29 +08:00
|
|
|
|
|
2023-02-08 04:27:55 +08:00
|
|
|
|
```javascript title="binary_tree.js"
|
2023-04-09 04:32:17 +08:00
|
|
|
|
/* 插入与删除节点 */
|
2022-12-04 12:26:09 +08:00
|
|
|
|
let P = new TreeNode(0);
|
2023-04-09 04:32:17 +08:00
|
|
|
|
// 在 n1 -> n2 中间插入节点 P
|
2022-12-04 12:26:09 +08:00
|
|
|
|
n1.left = P;
|
|
|
|
|
P.left = n2;
|
2023-04-09 04:32:17 +08:00
|
|
|
|
// 删除节点 P
|
2022-12-04 12:26:09 +08:00
|
|
|
|
n1.left = n2;
|
2022-12-03 01:31:29 +08:00
|
|
|
|
```
|
|
|
|
|
|
2023-07-26 15:35:38 +08:00
|
|
|
|
=== "TS"
|
2022-12-03 01:31:29 +08:00
|
|
|
|
|
|
|
|
|
```typescript title="binary_tree.ts"
|
2023-04-09 04:32:17 +08:00
|
|
|
|
/* 插入与删除节点 */
|
2022-12-15 00:00:40 +08:00
|
|
|
|
const P = new TreeNode(0);
|
2023-04-09 04:32:17 +08:00
|
|
|
|
// 在 n1 -> n2 中间插入节点 P
|
2022-12-15 00:00:40 +08:00
|
|
|
|
n1.left = P;
|
|
|
|
|
P.left = n2;
|
2023-04-09 04:32:17 +08:00
|
|
|
|
// 删除节点 P
|
2022-12-15 00:00:40 +08:00
|
|
|
|
n1.left = n2;
|
2022-12-03 01:31:29 +08:00
|
|
|
|
```
|
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
=== "Dart"
|
2022-12-03 01:31:29 +08:00
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
```dart title="binary_tree.dart"
|
2023-04-09 04:32:17 +08:00
|
|
|
|
/* 插入与删除节点 */
|
2022-12-23 15:42:02 +08:00
|
|
|
|
TreeNode P = new TreeNode(0);
|
2023-04-09 04:32:17 +08:00
|
|
|
|
// 在 n1 -> n2 中间插入节点 P
|
2022-12-23 15:42:02 +08:00
|
|
|
|
n1.left = P;
|
|
|
|
|
P.left = n2;
|
2023-04-09 04:32:17 +08:00
|
|
|
|
// 删除节点 P
|
2022-12-23 15:42:02 +08:00
|
|
|
|
n1.left = n2;
|
2022-12-03 01:31:29 +08:00
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```
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2023-09-04 03:19:08 +08:00
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=== "Rust"
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2023-02-01 22:03:04 +08:00
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2023-09-04 03:19:08 +08:00
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```rust title="binary_tree.rs"
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2023-10-01 19:25:03 +08:00
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let p = TreeNode::new(0);
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// 在 n1 -> n2 中间插入节点 P
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n1.borrow_mut().left = Some(p.clone());
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p.borrow_mut().left = Some(n2.clone());
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// 删除节点 p
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n1.borrow_mut().left = Some(n2);
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2023-02-01 22:03:04 +08:00
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```
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2023-09-04 03:19:08 +08:00
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=== "C"
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2023-06-02 02:40:26 +08:00
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2023-09-04 03:19:08 +08:00
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```c title="binary_tree.c"
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2023-06-02 14:56:29 +08:00
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/* 插入与删除节点 */
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2023-09-04 03:19:08 +08:00
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TreeNode *P = newTreeNode(0);
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2023-06-02 14:56:29 +08:00
|
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// 在 n1 -> n2 中间插入节点 P
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2023-09-04 03:19:08 +08:00
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n1->left = P;
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P->left = n2;
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2023-06-02 14:56:29 +08:00
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// 删除节点 P
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2023-09-04 03:19:08 +08:00
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n1->left = n2;
|
2023-06-02 02:40:26 +08:00
|
|
|
|
```
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|
2023-09-04 03:19:08 +08:00
|
|
|
|
=== "Zig"
|
2023-07-26 11:00:53 +08:00
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|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
```zig title="binary_tree.zig"
|
2023-07-26 11:00:53 +08:00
|
|
|
|
|
|
|
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|
```
|
|
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|
2022-11-22 17:47:26 +08:00
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|
!!! note
|
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|
2023-04-10 23:59:22 +08:00
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|
需要注意的是,插入节点可能会改变二叉树的原有逻辑结构,而删除节点通常意味着删除该节点及其所有子树。因此,在二叉树中,插入与删除操作通常是由一套操作配合完成的,以实现有实际意义的操作。
|
2022-11-22 17:47:26 +08:00
|
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|
|
|
2023-02-16 03:39:01 +08:00
|
|
|
|
## 常见二叉树类型
|
2022-11-22 17:47:26 +08:00
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|
|
|
2022-12-21 17:19:39 +08:00
|
|
|
|
### 完美二叉树
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2023-10-01 19:33:53 +08:00
|
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|
|
「完美二叉树 perfect binary tree」所有层的节点都被完全填满。在完美二叉树中,叶节点的度为 $0$ ,其余所有节点的度都为 $2$ ;若树高度为 $h$ ,则节点总数为 $2^{h+1} - 1$ ,呈现标准的指数级关系,反映了自然界中常见的细胞分裂现象。
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2022-12-21 17:19:39 +08:00
|
|
|
|
!!! tip
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2023-08-20 23:28:30 +08:00
|
|
|
|
请注意,在中文社区中,完美二叉树常被称为「满二叉树」。
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2023-02-26 18:18:34 +08:00
|
|
|
|
![完美二叉树](binary_tree.assets/perfect_binary_tree.png)
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2022-12-21 17:19:39 +08:00
|
|
|
|
### 完全二叉树
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2023-08-21 19:33:45 +08:00
|
|
|
|
如下图所示,「完全二叉树 complete binary tree」只有最底层的节点未被填满,且最底层节点尽量靠左填充。
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2023-02-26 18:18:34 +08:00
|
|
|
|
![完全二叉树](binary_tree.assets/complete_binary_tree.png)
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2022-12-21 17:19:39 +08:00
|
|
|
|
### 完满二叉树
|
2022-11-29 02:21:49 +08:00
|
|
|
|
|
2023-08-21 19:33:45 +08:00
|
|
|
|
如下图所示,「完满二叉树 full binary tree」除了叶节点之外,其余所有节点都有两个子节点。
|
2022-11-29 02:21:49 +08:00
|
|
|
|
|
2023-02-26 18:18:34 +08:00
|
|
|
|
![完满二叉树](binary_tree.assets/full_binary_tree.png)
|
2022-11-29 02:21:49 +08:00
|
|
|
|
|
2022-12-21 17:19:39 +08:00
|
|
|
|
### 平衡二叉树
|
2022-11-29 02:21:49 +08:00
|
|
|
|
|
2023-08-21 19:33:45 +08:00
|
|
|
|
如下图所示,「平衡二叉树 balanced binary tree」中任意节点的左子树和右子树的高度之差的绝对值不超过 1 。
|
2022-11-29 02:21:49 +08:00
|
|
|
|
|
2023-02-26 18:18:34 +08:00
|
|
|
|
![平衡二叉树](binary_tree.assets/balanced_binary_tree.png)
|
2022-11-29 02:21:49 +08:00
|
|
|
|
|
2023-02-16 03:39:01 +08:00
|
|
|
|
## 二叉树的退化
|
2022-12-03 01:31:29 +08:00
|
|
|
|
|
2023-10-01 19:33:53 +08:00
|
|
|
|
下图展示了二叉树的理想与退化状态。当二叉树的每层节点都被填满时,达到“完美二叉树”;而当所有节点都偏向一侧时,二叉树退化为“链表”。
|
2022-12-03 01:31:29 +08:00
|
|
|
|
|
2023-07-26 08:59:36 +08:00
|
|
|
|
- 完美二叉树是理想情况,可以充分发挥二叉树“分治”的优势。
|
|
|
|
|
- 链表则是另一个极端,各项操作都变为线性操作,时间复杂度退化至 $O(n)$ 。
|
2022-12-03 01:31:29 +08:00
|
|
|
|
|
2023-06-17 00:45:41 +08:00
|
|
|
|
![二叉树的最佳与最差结构](binary_tree.assets/binary_tree_best_worst_cases.png)
|
2022-12-03 01:31:29 +08:00
|
|
|
|
|
2023-04-09 04:32:17 +08:00
|
|
|
|
如下表所示,在最佳和最差结构下,二叉树的叶节点数量、节点总数、高度等达到极大或极小值。
|
2023-08-20 23:28:30 +08:00
|
|
|
|
|
2023-08-22 13:51:25 +08:00
|
|
|
|
<p align="center"> 表 <id> 二叉树的最佳与最差情况 </p>
|
2022-12-03 01:31:29 +08:00
|
|
|
|
|
2022-12-21 17:19:39 +08:00
|
|
|
|
| | 完美二叉树 | 链表 |
|
|
|
|
|
| ----------------------------- | ---------- | ---------- |
|
2023-04-09 04:32:17 +08:00
|
|
|
|
| 第 $i$ 层的节点数量 | $2^{i-1}$ | $1$ |
|
2023-08-19 19:22:08 +08:00
|
|
|
|
| 高度 $h$ 树的叶节点数量 | $2^h$ | $1$ |
|
|
|
|
|
| 高度 $h$ 树的节点总数 | $2^{h+1} - 1$ | $h + 1$ |
|
|
|
|
|
| 节点总数 $n$ 树的高度 | $\log_2 (n+1) - 1$ | $n - 1$ |
|