1. Add array representation of binary tree.

2. Update the LEGO example.
3. Update headers of the Python files.
This commit is contained in:
Yudong Jin 2022-12-23 00:44:40 +08:00
parent 9015222a0b
commit 919236f479
36 changed files with 151 additions and 61 deletions

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''' """
File: array.py File: array.py
Created Time: 2022-11-25 Created Time: 2022-11-25
Author: Krahets (krahets@163.com) Author: Krahets (krahets@163.com)
''' """
import sys, os.path as osp import sys, os.path as osp
sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))

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''' """
File: linked_list.py File: linked_list.py
Created Time: 2022-11-25 Created Time: 2022-11-25
Author: Krahets (krahets@163.com) Author: Krahets (krahets@163.com)
''' """
import sys, os.path as osp import sys, os.path as osp
sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))

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''' """
File: list.py File: list.py
Created Time: 2022-11-25 Created Time: 2022-11-25
Author: Krahets (krahets@163.com) Author: Krahets (krahets@163.com)
''' """
import sys, os.path as osp import sys, os.path as osp
sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))

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''' """
File: my_list.py File: my_list.py
Created Time: 2022-11-25 Created Time: 2022-11-25
Author: Krahets (krahets@163.com) Author: Krahets (krahets@163.com)
''' """
import sys, os.path as osp import sys, os.path as osp
sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))

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''' """
File: leetcode_two_sum.py File: leetcode_two_sum.py
Created Time: 2022-11-25 Created Time: 2022-11-25
Author: Krahets (krahets@163.com) Author: Krahets (krahets@163.com)
''' """
import sys, os.path as osp import sys, os.path as osp
sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))

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''' """
File: space_complexity.py File: space_complexity.py
Created Time: 2022-11-25 Created Time: 2022-11-25
Author: Krahets (krahets@163.com) Author: Krahets (krahets@163.com)
''' """
import sys, os.path as osp import sys, os.path as osp
sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))

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''' """
File: time_complexity.py File: time_complexity.py
Created Time: 2022-11-25 Created Time: 2022-11-25
Author: Krahets (krahets@163.com) Author: Krahets (krahets@163.com)
''' """
import sys, os.path as osp import sys, os.path as osp
sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))

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''' """
File: worst_best_time_complexity.py File: worst_best_time_complexity.py
Created Time: 2022-11-25 Created Time: 2022-11-25
Author: Krahets (krahets@163.com) Author: Krahets (krahets@163.com)
''' """
import sys, os.path as osp import sys, os.path as osp
sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))

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''' """
File: binary_search.py File: binary_search.py
Created Time: 2022-11-26 Created Time: 2022-11-26
Author: timi (xisunyy@163.com) Author: timi (xisunyy@163.com)
''' """
import sys, os.path as osp import sys, os.path as osp
sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))

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''' """
File: hashing_search.py File: hashing_search.py
Created Time: 2022-11-26 Created Time: 2022-11-26
Author: timi (xisunyy@163.com) Author: timi (xisunyy@163.com)
''' """
import sys, os.path as osp import sys, os.path as osp
sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))

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''' """
File: linear_search.py File: linear_search.py
Created Time: 2022-11-26 Created Time: 2022-11-26
Author: timi (xisunyy@163.com) Author: timi (xisunyy@163.com)
''' """
import sys, os.path as osp import sys, os.path as osp
sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))

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''' """
File: bubble_sort.py File: bubble_sort.py
Created Time: 2022-11-25 Created Time: 2022-11-25
Author: timi (xisunyy@163.com) Author: timi (xisunyy@163.com)
''' """
import sys, os.path as osp import sys, os.path as osp
sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))

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''' """
File: insertion_sort.py File: insertion_sort.py
Created Time: 2022-11-25 Created Time: 2022-11-25
Author: timi (xisunyy@163.com) Author: timi (xisunyy@163.com)
''' """
import sys, os.path as osp import sys, os.path as osp
sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))

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''' """
File: merge_sort.py File: merge_sort.py
Created Time: 2022-11-25 Created Time: 2022-11-25
Author: timi (xisunyy@163.com) Author: timi (xisunyy@163.com)
''' """
import sys, os.path as osp import sys, os.path as osp
sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))

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''' """
File: quick_sort.py File: quick_sort.py
Created Time: 2022-11-25 Created Time: 2022-11-25
Author: timi (xisunyy@163.com) Author: timi (xisunyy@163.com)
''' """
import sys, os.path as osp import sys, os.path as osp
sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))

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''' """
File: array_queue.py File: array_queue.py
Created Time: 2022-12-01 Created Time: 2022-12-01
Author: Peng Chen (pengchzn@gmail.com) Author: Peng Chen (pengchzn@gmail.com)
''' """
import os.path as osp import os.path as osp
import sys import sys

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''' """
File: array_stack.py File: array_stack.py
Created Time: 2022-11-29 Created Time: 2022-11-29
Author: Peng Chen (pengchzn@gmail.com) Author: Peng Chen (pengchzn@gmail.com)
''' """
import sys, os.path as osp import sys, os.path as osp
sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))

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''' """
File: deque.py File: deque.py
Created Time: 2022-11-29 Created Time: 2022-11-29
Author: Peng Chen (pengchzn@gmail.com) Author: Peng Chen (pengchzn@gmail.com)
''' """
import os.path as osp import os.path as osp
import sys import sys

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''' """
File: linkedlist_queue.py File: linkedlist_queue.py
Created Time: 2022-12-01 Created Time: 2022-12-01
Author: Peng Chen (pengchzn@gmail.com) Author: Peng Chen (pengchzn@gmail.com)
''' """
import os.path as osp import os.path as osp
import sys import sys

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''' """
File: linkedlist_stack.py File: linkedlist_stack.py
Created Time: 2022-11-29 Created Time: 2022-11-29
Author: Peng Chen (pengchzn@gmail.com) Author: Peng Chen (pengchzn@gmail.com)
''' """
import sys, os.path as osp import sys, os.path as osp
sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))

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''' """
File: queue.py File: queue.py
Created Time: 2022-11-29 Created Time: 2022-11-29
Author: Peng Chen (pengchzn@gmail.com) Author: Peng Chen (pengchzn@gmail.com)
''' """
import os.path as osp import os.path as osp
import sys import sys

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''' """
File: stack.py File: stack.py
Created Time: 2022-11-29 Created Time: 2022-11-29
Author: Peng Chen (pengchzn@gmail.com) Author: Peng Chen (pengchzn@gmail.com)
''' """
import sys, os.path as osp import sys, os.path as osp
sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))

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''' """
File: binary_search_tree.py File: binary_search_tree.py
Created Time: 2022-11-25 Created Time: 2022-11-25
Author: Krahets (krahets@163.com) Author: Krahets (krahets@163.com)
''' """
import sys, os.path as osp import sys, os.path as osp
sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))

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''' """
File: binary_tree.py File: binary_tree.py
Created Time: 2022-11-25 Created Time: 2022-11-25
Author: Krahets (krahets@163.com) Author: Krahets (krahets@163.com)
''' """
import sys, os.path as osp import sys, os.path as osp
sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))

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''' """
File: binary_tree_bfs.py File: binary_tree_bfs.py
Created Time: 2022-11-25 Created Time: 2022-11-25
Author: Krahets (krahets@163.com) Author: Krahets (krahets@163.com)
''' """
import sys, os.path as osp import sys, os.path as osp
sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))

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''' """
File: binary_tree_dfs.py File: binary_tree_dfs.py
Created Time: 2022-11-25 Created Time: 2022-11-25
Author: Krahets (krahets@163.com) Author: Krahets (krahets@163.com)
''' """
import sys, os.path as osp import sys, os.path as osp
sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))

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''' """
File: binary_tree.py File: binary_tree.py
Created Time: 2021-12-11 Created Time: 2021-12-11
Author: Krahets (krahets@163.com) Author: Krahets (krahets@163.com)
''' """
import collections import collections

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''' """
File: linked_list.py File: linked_list.py
Created Time: 2021-12-11 Created Time: 2021-12-11
Author: Krahets (krahets@163.com) Author: Krahets (krahets@163.com)
''' """
class ListNode: class ListNode:
"""Definition for a singly-linked list node """Definition for a singly-linked list node

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''' """
File: print_util.py File: print_util.py
Created Time: 2021-12-11 Created Time: 2021-12-11
Author: Krahets (krahets@163.com), msk397 (machangxinq@gmail.com) Author: Krahets (krahets@163.com), msk397 (machangxinq@gmail.com)
''' """
import copy import copy
import queue import queue

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@ -31,12 +31,23 @@ comments: true
- 算法是发挥数据结构优势的舞台。数据结构仅存储数据信息,结合算法才可解决特定问题。 - 算法是发挥数据结构优势的舞台。数据结构仅存储数据信息,结合算法才可解决特定问题。
- 算法有对应最优的数据结构。给定算法,一般可基于不同的数据结构实现,而最终执行效率往往相差很大。 - 算法有对应最优的数据结构。给定算法,一般可基于不同的数据结构实现,而最终执行效率往往相差很大。
如果将数据结构与算法比作「LEGO 乐高」,数据结构就是乐高「积木」,而算法就是把积木拼成目标形态的一系列「操作步骤」。
![relationship_between_data_structure_and_algorithm](what_is_dsa.assets/relationship_between_data_structure_and_algorithm.png) ![relationship_between_data_structure_and_algorithm](what_is_dsa.assets/relationship_between_data_structure_and_algorithm.png)
<p align="center"> Fig. 数据结构与算法的关系 </p> <p align="center"> Fig. 数据结构与算法的关系 </p>
如果将「LEGO 乐高」类比到「数据结构与算法」,那么可以得到下表所示的对应关系。
<div class="center-table" markdown>
| 数据结构与算法 | LEGO 乐高 |
| -------------- | ---------------------------------------- |
| 输入数据 | 未拼装的积木 |
| 数据结构 | 积木组织形式,包括形状、大小、连接方式等 |
| 算法 | 把积木拼成目标形态的一系列操作步骤 |
| 输出数据 | 积木模型 |
</div>
!!! tip "约定俗成的简称" !!! tip "约定俗成的简称"
在实际讨论中,我们通常会将「数据结构与算法」直接简称为「算法」。例如,我们熟称的 LeetCode 算法题目,实际上同时考察了数据结构和算法两部分知识。 在实际讨论中,我们通常会将「数据结构与算法」直接简称为「算法」。例如,我们熟称的 LeetCode 算法题目,实际上同时考察了数据结构和算法两部分知识。

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| 树的结点总数为 $n$ 时的高度 | $\log_2 (n+1) - 1$ | $n - 1$ | | 树的结点总数为 $n$ 时的高度 | $\log_2 (n+1) - 1$ | $n - 1$ |
</div> </div>
## 二叉树表示方式 *
我们一般使用二叉树的「链表表示」,即存储单位为结点 `TreeNode` ,结点之间通过指针(引用)相连接。本文前述示例代码展示了二叉树在链表表示下的各项基本操作。
那能否可以用「数组表示」二叉树呢?答案是肯定的。先来分析一个简单案例,给定一个「完美二叉树」,将结点按照层序遍历的顺序编号(从 0 开始),那么可以推导得出父结点索引与子结点索引之间的「映射公式」:**设结点的索引为 $i$ ,则该结点的左子结点索引为 $2i + 1$ 、右子结点索引为 $2i + 2$** 。
**本质上,映射公式的作用就是链表中的指针**。对于层序遍历序列中的任意结点,我们都可以使用映射公式来访问子结点。因此,可以直接使用层序遍历序列(即数组)来表示完美二叉树。
![array_representation_mapping](binary_tree.assets/array_representation_mapping.png)
然而,完美二叉树只是个例,二叉树中间层往往存在许多空结点(即 `null` ),而层序遍历序列并不包含这些空结点,并且我们无法单凭序列来猜测空结点的数量和分布位置,**即理论上存在许多种二叉树都符合该层序遍历序列**。显然,这种情况无法使用数组来存储二叉树。
![array_representation_without_empty](binary_tree.assets/array_representation_without_empty.png)
为了解决此问题,考虑按照完美二叉树的形式来表示所有二叉树,**即在序列中使用特殊符号来显式地表示“空位”**。如下图所示,这样处理后,序列(数组)就可以唯一表示二叉树了。
=== "Java"
```java title=""
/* 二叉树的数组表示 */
// 使用 int 的包装类 Integer ,就可以使用 null 来标记空位
Integer[] tree = { 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 };
```
=== "C++"
```cpp title=""
/* 二叉树的数组表示 */
// 为了符合数据类型为 int ,使用 int 最大值标记空位
// 该方法的使用前提是没有结点的值 = INT_MAX
vector<int> tree = { 1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15 };
```
=== "Python"
```python title=""
“”“ 二叉树的数组表示 ”“”
# 直接使用 None 来表示空位
tree = [1, 2, 3, 4, None, 6, 7, 8, 9, None, None, 12, None, None, 15]
```
=== "Go"
```go title=""
```
=== "JavaScript"
```js title=""
```
=== "TypeScript"
```typescript title=""
```
=== "C"
```c title=""
```
=== "C#"
```csharp title=""
```
![array_representation_with_empty](binary_tree.assets/array_representation_with_empty.png)
回顾「完全二叉树」的满足条件,其只有最底层有空结点,并且最底层的结点尽量靠左,因而所有空结点都一定出现在层序遍历序列的末尾。**因为我们先验地确定了空位的位置,所以在使用数组表示完全二叉树时,可以省略存储“空位”**。“便于使用数组表示”也是完全二叉树受欢迎的原因之一。
![array_representation_complete_binary_tree](binary_tree.assets/array_representation_complete_binary_tree.png)
数组表示有两个优点: 一是不需要存储指针,节省空间;二是可以随机访问结点。然而,当二叉树中的“空位”很多时,数组中只包含很少结点的数据,空间利用率很低。