# 2.1 - Tree

### 1. 定義

Tree是由n個節點(node)所構成之集合，不可為空。

1. 至少有一個Root
2. 其餘的節點可分為T1\~Tm個互斥集合，稱為Root的子樹(subtree)。

### 2. 術語

```
                   Level
       A          -  0(1)
    / |  \ 
   B  E   G       -  1(2)
 / |  | / | \ 
 C D  F H I  J    -  2(3)
```

1. **Degree of nodes**
   1. Degree of A = 3 
   2. Degreee of B = 2
2. **Leaf**: Degree = 0
3. **Non-leaf**: A, B, E, G
4. **Child**: B is a child of A
5. **Parent**: A is the parent of B
6. **Sibling**: 有相同Parent彼此稱之
7. **Ancestors & Descendants**
8. **Level of nodes**
9. **Degree of a Tree** = Max(degrees of nodes)
10. **Height of a Tree** = Max(levels of nodes)
11. **Forest**: >=0個互斥的樹所形成之集合。

### 3. 表示法

#### 1. 使用Link List表示

![](https://2769815795-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LGckN3OAfinKuVIrkMj%2F-LHGZeAGCU8T4to40ViB%2F-LHGa7I8w9S5O8Pwz8Dq%2F%E8%9E%A2%E5%B9%95%E5%BF%AB%E7%85%A7%202018-07-13%2011.03.00.jpg?alt=media\&token=9cb22fb2-7b9a-4559-aaae-09886ec0ebf2)

Nodes = n , Degree of tree = k

Available links = n-1（扣掉root，其他節點都有一個有用的Link）

Empty links = n\*k - (n-1)

$$
\frac{n*k - (n-1)  }{n*k} = \frac{k-1}{k}
$$

非常浪費空間的表示法。

#### 2. 使用Binary Tree表示。

#### 3. Sibling-Child表示法。

1. 節點中的child指標指向leftmost-child(A->B)
2. 節點中的sibling指標指向right-sibling(B->C, C->D)

![](https://2769815795-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LGckN3OAfinKuVIrkMj%2F-LHGZeAGCU8T4to40ViB%2F-LHGjUL0Hess4Y6Gy634%2F%E8%9E%A2%E5%B9%95%E5%BF%AB%E7%85%A7%202018-07-13%2011.44.39.jpg?alt=media\&token=adcdea58-300e-4d62-985d-ea9611339568)

#### 4. 括號法

```
       A    
    / |  \ 
   B  E   G 
 / |  | / | \ 
 C D  F H I  J  
```

Ａ（Ｂ（ＣＤ）Ｅ（Ｆ）Ｇ（ＨＩＪ））


---

# Agent Instructions: Querying This Documentation

If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

```
GET https://garychang.gitbook.io/data-structure/lecture2-tree-and-binary-tree/lecture2.1-tree.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.