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hello-algo/zh-hant/docs/chapter_tree/binary_tree.md
Yudong Jin 5849ae4ada
Sync zh and zh-hant versions. (#1523)
2024-09-28 09:28:44 +08:00

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# 二元樹
<u>二元樹（binary tree）</u>是一種非線性資料結構，代表“祖先”與“後代”之間的派生關係，體現了“一分為二”的分治邏輯。與鏈結串列類似，二元樹的基本單元是節點，每個節點包含值、左子節點引用和右子節點引用。
=== "Python"
```python title=""
class TreeNode:
"""二元樹節點類別"""
def __init__(self, val: int):
self.val: int = val # 節點值
self.left: TreeNode | None = None # 左子節點引用
self.right: TreeNode | None = None # 右子節點引用
```
=== "C++"
```cpp title=""
/* 二元樹節點結構體 */
struct TreeNode {
int val; // 節點值
TreeNode *left; // 左子節點指標
TreeNode *right; // 右子節點指標
TreeNode(int x) : val(x), left(nullptr), right(nullptr) {}
};
```
=== "Java"
```java title=""
/* 二元樹節點類別 */
class TreeNode {
int val; // 節點值
TreeNode left; // 左子節點引用
TreeNode right; // 右子節點引用
TreeNode(int x) { val = x; }
}
```
=== "C#"
```csharp title=""
/* 二元樹節點類別 */
class TreeNode(int? x) {
public int? val = x; // 節點值
public TreeNode? left; // 左子節點引用
public TreeNode? right; // 右子節點引用
}
```
=== "Go"
```go title=""
/* 二元樹節點結構體 */
type TreeNode struct {
Val int
Left *TreeNode
Right *TreeNode
}
/* 建構子 */
func NewTreeNode(v int) *TreeNode {
return &TreeNode{
Left: nil, // 左子節點指標
Right: nil, // 右子節點指標
Val: v, // 節點值
}
}
```
=== "Swift"
```swift title=""
/* 二元樹節點類別 */
class TreeNode {
var val: Int // 節點值
var left: TreeNode? // 左子節點引用
var right: TreeNode? // 右子節點引用
init(x: Int) {
val = x
}
}
```
=== "JS"
```javascript title=""
/* 二元樹節點類別 */
class TreeNode {
val; // 節點值
left; // 左子節點指標
right; // 右子節點指標
constructor(val, left, right) {
this.val = val === undefined ? 0 : val;
this.left = left === undefined ? null : left;
this.right = right === undefined ? null : right;
}
}
```
=== "TS"
```typescript title=""
/* 二元樹節點類別 */
class TreeNode {
val: number;
left: TreeNode | null;
right: TreeNode | null;
constructor(val?: number, left?: TreeNode | null, right?: TreeNode | null) {
this.val = val === undefined ? 0 : val; // 節點值
this.left = left === undefined ? null : left; // 左子節點引用
this.right = right === undefined ? null : right; // 右子節點引用
}
}
```
=== "Dart"
```dart title=""
/* 二元樹節點類別 */
class TreeNode {
int val; // 節點值
TreeNode? left; // 左子節點引用
TreeNode? right; // 右子節點引用
TreeNode(this.val, [this.left, this.right]);
}
```
=== "Rust"
```rust title=""
use std::rc::Rc;
use std::cell::RefCell;
/* 二元樹節點結構體 */
struct TreeNode {
val: i32, // 節點值
left: Option<Rc<RefCell<TreeNode>>>, // 左子節點引用
right: Option<Rc<RefCell<TreeNode>>>, // 右子節點引用
}
impl TreeNode {
/* 建構子 */
fn new(val: i32) -> Rc<RefCell<Self>> {
Rc::new(RefCell::new(Self {
val,
left: None,
right: None
}))
}
}
```
=== "C"
```c title=""
/* 二元樹節點結構體 */
typedef struct TreeNode {
int val; // 節點值
int height; // 節點高度
struct TreeNode *left; // 左子節點指標
struct TreeNode *right; // 右子節點指標
} TreeNode;
/* 建構子 */
TreeNode *newTreeNode(int val) {
TreeNode *node;
node = (TreeNode *)malloc(sizeof(TreeNode));
node->val = val;
node->height = 0;
node->left = NULL;
node->right = NULL;
return node;
}
```
=== "Kotlin"
```kotlin title=""
/* 二元樹節點類別 */
class TreeNode(val _val: Int) { // 節點值
val left: TreeNode? = null // 左子節點引用
val right: TreeNode? = null // 右子節點引用
}
```
=== "Ruby"
```ruby title=""
### 二元樹節點類別 ###
class TreeNode
attr_accessor :val # 節點值
attr_accessor :left # 左子節點引用
attr_accessor :right # 右子節點引用
def initialize(val)
@val = val
end
end
```
=== "Zig"
```zig title=""
```
每個節點都有兩個引用（指標），分別指向<u>左子節點（left-child node）</u>和<u>右子節點（right-child node）</u>，該節點被稱為這兩個子節點的<u>父節點（parent node）</u>。當給定一個二元樹的節點時，我們將該節點的左子節點及其以下節點形成的樹稱為該節點的<u>左子樹（left subtree）</u>，同理可得<u>右子樹（right subtree）</u>。
**在二元樹中，除葉節點外，其他所有節點都包含子節點和非空子樹**。如下圖所示，如果將“節點 2”視為父節點，則其左子節點和右子節點分別是“節點 4”和“節點 5”，左子樹是“節點 4 及其以下節點形成的樹”，右子樹是“節點 5 及其以下節點形成的樹”。
![父節點、子節點、子樹](binary_tree.assets/binary_tree_definition.png)
## 二元樹常見術語
二元樹的常用術語如下圖所示。
- <u>根節點（root node）</u>：位於二元樹頂層的節點，沒有父節點。
- <u>葉節點（leaf node）</u>：沒有子節點的節點，其兩個指標均指向 `None` 。
- <u>邊（edge）</u>：連線兩個節點的線段，即節點引用（指標）。
- 節點所在的<u>層（level）</u>：從頂至底遞增，根節點所在層為 1 。
- 節點的<u>度（degree）</u>：節點的子節點的數量。在二元樹中，度的取值範圍是 0、1、2 。
- 二元樹的<u>高度（height）</u>：從根節點到最遠葉節點所經過的邊的數量。
- 節點的<u>深度（depth）</u>：從根節點到該節點所經過的邊的數量。
- 節點的<u>高度（height）</u>：從距離該節點最遠的葉節點到該節點所經過的邊的數量。
![二元樹的常用術語](binary_tree.assets/binary_tree_terminology.png)
!!! tip
請注意，我們通常將“高度”和“深度”定義為“經過的邊的數量”，但有些題目或教材可能會將其定義為“經過的節點的數量”。在這種情況下，高度和深度都需要加 1 。
## 二元樹基本操作
### 初始化二元樹
與鏈結串列類似，首先初始化節點，然後構建引用（指標）。
=== "Python"
```python title="binary_tree.py"
# 初始化二元樹
# 初始化節點
n1 = TreeNode(val=1)
n2 = TreeNode(val=2)
n3 = TreeNode(val=3)
n4 = TreeNode(val=4)
n5 = TreeNode(val=5)
# 構建節點之間的引用（指標）
n1.left = n2
n1.right = n3
n2.left = n4
n2.right = n5
```
=== "C++"
```cpp title="binary_tree.cpp"
/* 初始化二元樹 */
// 初始化節點
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);
// 構建節點之間的引用（指標）
n1->left = n2;
n1->right = n3;
n2->left = n4;
n2->right = n5;
```
=== "Java"
```java title="binary_tree.java"
// 初始化節點
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);
// 構建節點之間的引用（指標）
n1.left = n2;
n1.right = n3;
n2.left = n4;
n2.right = n5;
```
=== "C#"
```csharp title="binary_tree.cs"
/* 初始化二元樹 */
// 初始化節點
TreeNode n1 = new(1);
TreeNode n2 = new(2);
TreeNode n3 = new(3);
TreeNode n4 = new(4);
TreeNode n5 = new(5);
// 構建節點之間的引用（指標）
n1.left = n2;
n1.right = n3;
n2.left = n4;
n2.right = n5;
```
=== "Go"
```go title="binary_tree.go"
/* 初始化二元樹 */
// 初始化節點
n1 := NewTreeNode(1)
n2 := NewTreeNode(2)
n3 := NewTreeNode(3)
n4 := NewTreeNode(4)
n5 := NewTreeNode(5)
// 構建節點之間的引用（指標）
n1.Left = n2
n1.Right = n3
n2.Left = n4
n2.Right = n5
```
=== "Swift"
```swift title="binary_tree.swift"
// 初始化節點
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)
// 構建節點之間的引用（指標）
n1.left = n2
n1.right = n3
n2.left = n4
n2.right = n5
```
=== "JS"
```javascript title="binary_tree.js"
/* 初始化二元樹 */
// 初始化節點
let n1 = new TreeNode(1),
n2 = new TreeNode(2),
n3 = new TreeNode(3),
n4 = new TreeNode(4),
n5 = new TreeNode(5);
// 構建節點之間的引用（指標）
n1.left = n2;
n1.right = n3;
n2.left = n4;
n2.right = n5;
```
=== "TS"
```typescript title="binary_tree.ts"
/* 初始化二元樹 */
// 初始化節點
let n1 = new TreeNode(1),
n2 = new TreeNode(2),
n3 = new TreeNode(3),
n4 = new TreeNode(4),
n5 = new TreeNode(5);
// 構建節點之間的引用（指標）
n1.left = n2;
n1.right = n3;
n2.left = n4;
n2.right = n5;
```
=== "Dart"
```dart title="binary_tree.dart"
/* 初始化二元樹 */
// 初始化節點
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);
// 構建節點之間的引用（指標）
n1.left = n2;
n1.right = n3;
n2.left = n4;
n2.right = n5;
```
=== "Rust"
```rust title="binary_tree.rs"
// 初始化節點
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);
// 構建節點之間的引用（指標）
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);
```
=== "C"
```c title="binary_tree.c"
/* 初始化二元樹 */
// 初始化節點
TreeNode *n1 = newTreeNode(1);
TreeNode *n2 = newTreeNode(2);
TreeNode *n3 = newTreeNode(3);
TreeNode *n4 = newTreeNode(4);
TreeNode *n5 = newTreeNode(5);
// 構建節點之間的引用（指標）
n1->left = n2;
n1->right = n3;
n2->left = n4;
n2->right = n5;
```
=== "Kotlin"
```kotlin title="binary_tree.kt"
// 初始化節點
val n1 = TreeNode(1)
val n2 = TreeNode(2)
val n3 = TreeNode(3)
val n4 = TreeNode(4)
val n5 = TreeNode(5)
// 構建節點之間的引用（指標）
n1.left = n2
n1.right = n3
n2.left = n4
n2.right = n5
```
=== "Ruby"
```ruby title="binary_tree.rb"
# 初始化二元樹
# 初始化節點
n1 = TreeNode.new(1)
n2 = TreeNode.new(2)
n3 = TreeNode.new(3)
n4 = TreeNode.new(4)
n5 = TreeNode.new(5)
# 構建節點之間的引用（指標）
n1.left = n2
n1.right = n3
n2.left = n4
n2.right = n5
```
=== "Zig"
```zig title="binary_tree.zig"
```
??? pythontutor "視覺化執行"
https://pythontutor.com/render.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%85%83%E6%A8%B9%E7%AF%80%E9%BB%9E%E9%A1%9E%E5%88%A5%22%22%22%0A%20%20%20%20def%20__init__%28self%2C%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E7%AF%80%E9%BB%9E%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E7%AF%80%E9%BB%9E%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20%E5%8F%B3%E5%AD%90%E7%AF%80%E9%BB%9E%E5%BC%95%E7%94%A8%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E4%BA%8C%E5%85%83%E6%A8%B9%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E7%AF%80%E9%BB%9E%0A%20%20%20%20n1%20%3D%20TreeNode%28val%3D1%29%0A%20%20%20%20n2%20%3D%20TreeNode%28val%3D2%29%0A%20%20%20%20n3%20%3D%20TreeNode%28val%3D3%29%0A%20%20%20%20n4%20%3D%20TreeNode%28val%3D4%29%0A%20%20%20%20n5%20%3D%20TreeNode%28val%3D5%29%0A%20%20%20%20%23%20%E6%A7%8B%E5%BB%BA%E7%AF%80%E9%BB%9E%E4%B9%8B%E9%96%93%E7%9A%84%E5%BC%95%E7%94%A8%EF%BC%88%E6%8C%87%E6%A8%99%EF%BC%89%0A%20%20%20%20n1.left%20%3D%20n2%0A%20%20%20%20n1.right%20%3D%20n3%0A%20%20%20%20n2.left%20%3D%20n4%0A%20%20%20%20n2.right%20%3D%20n5&cumulative=false&curInstr=3&heapPrimitives=nevernest&mode=display&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false
### 插入與刪除節點
與鏈結串列類似，在二元樹中插入與刪除節點可以透過修改指標來實現。下圖給出了一個示例。
![在二元樹中插入與刪除節點](binary_tree.assets/binary_tree_add_remove.png)
=== "Python"
```python title="binary_tree.py"
# 插入與刪除節點
p = TreeNode(0)
# 在 n1 -> n2 中間插入節點 P
n1.left = p
p.left = n2
# 刪除節點 P
n1.left = n2
```
=== "C++"
```cpp title="binary_tree.cpp"
/* 插入與刪除節點 */
TreeNode* P = new TreeNode(0);
// 在 n1 -> n2 中間插入節點 P
n1->left = P;
P->left = n2;
// 刪除節點 P
n1->left = n2;
```
=== "Java"
```java title="binary_tree.java"
TreeNode P = new TreeNode(0);
// 在 n1 -> n2 中間插入節點 P
n1.left = P;
P.left = n2;
// 刪除節點 P
n1.left = n2;
```
=== "C#"
```csharp title="binary_tree.cs"
/* 插入與刪除節點 */
TreeNode P = new(0);
// 在 n1 -> n2 中間插入節點 P
n1.left = P;
P.left = n2;
// 刪除節點 P
n1.left = n2;
```
=== "Go"
```go title="binary_tree.go"
/* 插入與刪除節點 */
// 在 n1 -> n2 中間插入節點 P
p := NewTreeNode(0)
n1.Left = p
p.Left = n2
// 刪除節點 P
n1.Left = n2
```
=== "Swift"
```swift title="binary_tree.swift"
let P = TreeNode(x: 0)
// 在 n1 -> n2 中間插入節點 P
n1.left = P
P.left = n2
// 刪除節點 P
n1.left = n2
```
=== "JS"
```javascript title="binary_tree.js"
/* 插入與刪除節點 */
let P = new TreeNode(0);
// 在 n1 -> n2 中間插入節點 P
n1.left = P;
P.left = n2;
// 刪除節點 P
n1.left = n2;
```
=== "TS"
```typescript title="binary_tree.ts"
/* 插入與刪除節點 */
const P = new TreeNode(0);
// 在 n1 -> n2 中間插入節點 P
n1.left = P;
P.left = n2;
// 刪除節點 P
n1.left = n2;
```
=== "Dart"
```dart title="binary_tree.dart"
/* 插入與刪除節點 */
TreeNode P = new TreeNode(0);
// 在 n1 -> n2 中間插入節點 P
n1.left = P;
P.left = n2;
// 刪除節點 P
n1.left = n2;
```
=== "Rust"
```rust title="binary_tree.rs"
let p = TreeNode::new(0);
// 在 n1 -> n2 中間插入節點 P
n1.borrow_mut().left = Some(p.clone());
p.borrow_mut().left = Some(n2.clone());
// 刪除節點 p
n1.borrow_mut().left = Some(n2);
```
=== "C"
```c title="binary_tree.c"
/* 插入與刪除節點 */
TreeNode *P = newTreeNode(0);
// 在 n1 -> n2 中間插入節點 P
n1->left = P;
P->left = n2;
// 刪除節點 P
n1->left = n2;
```
=== "Kotlin"
```kotlin title="binary_tree.kt"
val P = TreeNode(0)
// 在 n1 -> n2 中間插入節點 P
n1.left = P
P.left = n2
// 刪除節點 P
n1.left = n2
```
=== "Ruby"
```ruby title="binary_tree.rb"
# 插入與刪除節點
_p = TreeNode.new(0)
# 在 n1 -> n2 中間插入節點 _p
n1.left = _p
_p.left = n2
# 刪除節點
n1.left = n2
```
=== "Zig"
```zig title="binary_tree.zig"
```
??? pythontutor "視覺化執行"
https://pythontutor.com/render.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%85%83%E6%A8%B9%E7%AF%80%E9%BB%9E%E9%A1%9E%E5%88%A5%22%22%22%0A%20%20%20%20def%20__init__%28self%2C%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E7%AF%80%E9%BB%9E%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E7%AF%80%E9%BB%9E%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20%E5%8F%B3%E5%AD%90%E7%AF%80%E9%BB%9E%E5%BC%95%E7%94%A8%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E4%BA%8C%E5%85%83%E6%A8%B9%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E7%AF%80%E9%BB%9E%0A%20%20%20%20n1%20%3D%20TreeNode%28val%3D1%29%0A%20%20%20%20n2%20%3D%20TreeNode%28val%3D2%29%0A%20%20%20%20n3%20%3D%20TreeNode%28val%3D3%29%0A%20%20%20%20n4%20%3D%20TreeNode%28val%3D4%29%0A%20%20%20%20n5%20%3D%20TreeNode%28val%3D5%29%0A%20%20%20%20%23%20%E6%A7%8B%E5%BB%BA%E7%AF%80%E9%BB%9E%E4%B9%8B%E9%96%93%E7%9A%84%E5%BC%95%E7%94%A8%EF%BC%88%E6%8C%87%E6%A8%99%EF%BC%89%0A%20%20%20%20n1.left%20%3D%20n2%0A%20%20%20%20n1.right%20%3D%20n3%0A%20%20%20%20n2.left%20%3D%20n4%0A%20%20%20%20n2.right%20%3D%20n5%0A%0A%20%20%20%20%23%20%E6%8F%92%E5%85%A5%E8%88%87%E5%88%AA%E9%99%A4%E7%AF%80%E9%BB%9E%0A%20%20%20%20p%20%3D%20TreeNode%280%29%0A%20%20%20%20%23%20%E5%9C%A8%20n1%20-%3E%20n2%20%E4%B8%AD%E9%96%93%E6%8F%92%E5%85%A5%E7%AF%80%E9%BB%9E%20P%0A%20%20%20%20n1.left%20%3D%20p%0A%20%20%20%20p.left%20%3D%20n2%0A%20%20%20%20%23%20%E5%88%AA%E9%99%A4%E7%AF%80%E9%BB%9E%20P%0A%20%20%20%20n1.left%20%3D%20n2&cumulative=false&curInstr=37&heapPrimitives=nevernest&mode=display&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false
!!! tip
需要注意的是，插入節點可能會改變二元樹的原有邏輯結構，而刪除節點通常意味著刪除該節點及其所有子樹。因此，在二元樹中，插入與刪除通常是由一套操作配合完成的，以實現有實際意義的操作。
## 常見二元樹型別
### 完美二元樹
如下圖所示，<u>完美二元樹（perfect binary tree）</u>所有層的節點都被完全填滿。在完美二元樹中，葉節點的度為 $0$ ，其餘所有節點的度都為 $2$ ；若樹的高度為 $h$ ，則節點總數為 $2^{h+1} - 1$ ，呈現標準的指數級關係，反映了自然界中常見的細胞分裂現象。
!!! tip
請注意，在中文社群中，完美二元樹常被稱為<u>滿二元樹</u>。
![完美二元樹](binary_tree.assets/perfect_binary_tree.png)
### 完全二元樹
如下圖所示，<u>完全二元樹（complete binary tree）</u>只有最底層的節點未被填滿，且最底層節點儘量靠左填充。請注意，完美二元樹也是一棵完全二元樹。
![完全二元樹](binary_tree.assets/complete_binary_tree.png)
### 完滿二元樹
如下圖所示，<u>完滿二元樹（full binary tree）</u>除了葉節點之外，其餘所有節點都有兩個子節點。
![完滿二元樹](binary_tree.assets/full_binary_tree.png)
### 平衡二元樹
如下圖所示，<u>平衡二元樹（balanced binary tree）</u>中任意節點的左子樹和右子樹的高度之差的絕對值不超過 1 。
![平衡二元樹](binary_tree.assets/balanced_binary_tree.png)
## 二元樹的退化
下圖展示了二元樹的理想結構與退化結構。當二元樹的每層節點都被填滿時，達到“完美二元樹”；而當所有節點都偏向一側時，二元樹退化為“鏈結串列”。
- 完美二元樹是理想情況，可以充分發揮二元樹“分治”的優勢。
- 鏈結串列則是另一個極端，各項操作都變為線性操作，時間複雜度退化至 $O(n)$ 。
![二元樹的最佳結構與最差結構](binary_tree.assets/binary_tree_best_worst_cases.png)
如下表所示，在最佳結構和最差結構下，二元樹的葉節點數量、節點總數、高度等達到極大值或極小值。
<p align="center"> 表 <id> &nbsp; 二元樹的最佳結構與最差結構 </p>
| | 完美二元樹 | 鏈結串列 |
| --------------------------- | ------------------ | ------- |
| 第 $i$ 層的節點數量 | $2^{i-1}$ | $1$ |
| 高度為 $h$ 的樹的葉節點數量 | $2^h$ | $1$ |
| 高度為 $h$ 的樹的節點總數 | $2^{h+1} - 1$ | $h + 1$ |
| 節點總數為 $n$ 的樹的高度 | $\log_2 (n+1) - 1$ | $n - 1$ |
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