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feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
# 构建二叉树问题
!!! question
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
2023-12-02 06:21:34 +08:00
给定一棵二叉树的前序遍历 `preorder` 和中序遍历 `inorder` ，请从中构建二叉树，返回二叉树的根节点。假设二叉树中没有值重复的节点。
feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
Fix naming of the section build_binary_tree_problem
2023-07-17 02:18:53 +08:00
![构建二叉树的示例数据](build_binary_tree_problem.assets/build_tree_example.png)
feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
Add subtitles to docs
2023-07-21 21:54:51 +08:00
### 判断是否为分治问题
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
2023-12-02 06:21:34 +08:00
原问题定义为从 `preorder` 和 `inorder` 构建二叉树，是一个典型的分治问题。
feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
2023-12-02 06:21:34 +08:00
- **问题可以分解**：从分治的角度切入，我们可以将原问题划分为两个子问题：构建左子树、构建右子树，加上一步操作：初始化根节点。而对于每棵子树（子问题），我们仍然可以复用以上划分方法，将其划分为更小的子树（子问题），直至达到最小子问题（空子树）时终止。
- **子问题是独立的**：左子树和右子树是相互独立的，它们之间没有交集。在构建左子树时，我们只需关注中序遍历和前序遍历中与左子树对应的部分。右子树同理。
Finetune the chapter of hashing, divide and conquer, backtracking, tree
2023-07-24 03:04:55 +08:00
- **子问题的解可以合并**：一旦得到了左子树和右子树（子问题的解），我们就可以将它们链接到根节点上，得到原问题的解。
feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
Add subtitles to docs
2023-07-21 21:54:51 +08:00
### 如何划分子树
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
2023-12-02 06:21:34 +08:00
根据以上分析，这道题可以使用分治来求解，**但如何通过前序遍历 `preorder` 和中序遍历 `inorder` 来划分左子树和右子树呢**？
feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
2023-12-02 06:21:34 +08:00
根据定义，`preorder` 和 `inorder` 都可以划分为三个部分。
feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
Add figure and table numbers in normal texts.
2023-08-22 13:51:25 +08:00
- 前序遍历：`[ 根节点 | 左子树 | 右子树 ]` ，例如上图的树对应 `[ 3 | 9 | 2 1 7 ]` 。
- 中序遍历：`[ 左子树 | 根节点 ｜ 右子树 ]` ，例如上图的树对应 `[ 9 | 3 | 1 2 7 ]` 。
feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
Replace "：" with "。"
2023-08-27 22:49:47 +08:00
以上图数据为例，我们可以通过下图所示的步骤得到划分结果。
feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
Unify punctuation.
2023-07-26 08:59:36 +08:00
1. 前序遍历的首元素 3 是根节点的值。
2. 查找根节点 3 在 `inorder` 中的索引，利用该索引可将 `inorder` 划分为 `[ 9 | 3 ｜ 1 2 7 ]` 。
Update the book based on the revised second edition (#1014) * Revised the book * Update the book with the second revised edition * Revise base on the manuscript of the first edition
2023-12-28 18:06:09 +08:00
3. 根据 `inorder` 的划分结果，易得左子树和右子树的节点数量分别为 1 和 3 ，从而可将 `preorder` 划分为 `[ 3 | 9 | 2 1 7 ]` 。
feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
2023-12-02 06:21:34 +08:00
![在前序遍历和中序遍历中划分子树](build_binary_tree_problem.assets/build_tree_preorder_inorder_division.png)
feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
Finetune the chapter of hashing, divide and conquer, backtracking, tree
2023-07-24 03:04:55 +08:00
### 基于变量描述子树区间
Add subtitles to docs
2023-07-21 21:54:51 +08:00
Replace "：" with "。"
2023-08-27 22:49:47 +08:00
根据以上划分方法，**我们已经得到根节点、左子树、右子树在 `preorder` 和 `inorder` 中的索引区间**。而为了描述这些索引区间，我们需要借助几个指针变量。
feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
Unify punctuation.
2023-07-26 08:59:36 +08:00
- 将当前树的根节点在 `preorder` 中的索引记为 $i$ 。
- 将当前树的根节点在 `inorder` 中的索引记为 $m$ 。
- 将当前树在 `inorder` 中的索引区间记为 $[l, r]$ 。
feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
Finetune the chapter of hashing, divide and conquer, backtracking, tree
2023-07-24 03:04:55 +08:00
如下表所示，通过以上变量即可表示根节点在 `preorder` 中的索引，以及子树在 `inorder` 中的索引区间。
1. lower-case nouns 2. fix 2 figures 3. Replace some 「」 by “”
2023-08-20 23:28:30 +08:00
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
2023-12-02 06:21:34 +08:00
<p align="center"> 表 <id> &nbsp; 根节点和子树在前序遍历和中序遍历中的索引 </p>
feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
Finetune the chapter of hashing, divide and conquer, backtracking, tree
2023-07-24 03:04:55 +08:00
| | 根节点在 `preorder` 中的索引 | 子树在 `inorder` 中的索引区间 |
Format code and docs.
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| ------ | ---------------------------- | ----------------------------- |
| 当前树 | $i$ | $[l, r]$ |
| 左子树 | $i + 1$ | $[l, m-1]$ |
| 右子树 | $i + 1 + (m - l)$ | $[m+1, r]$ |
feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
Update the book based on the revised second edition (#1014) * Revised the book * Update the book with the second revised edition * Revise base on the manuscript of the first edition
2023-12-28 18:06:09 +08:00
请注意，右子树根节点索引中的 $(m-l)$ 的含义是“左子树的节点数量”，建议结合下图理解。
feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
Fix naming of the section build_binary_tree_problem
2023-07-17 02:18:53 +08:00
![根节点和左右子树的索引区间表示](build_binary_tree_problem.assets/build_tree_division_pointers.png)
feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
Add subtitles to docs
2023-07-21 21:54:51 +08:00
### 代码实现
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
2023-12-02 06:21:34 +08:00
为了提升查询 $m$ 的效率，我们借助一个哈希表 `hmap` 来存储数组 `inorder` 中元素到索引的映射：
feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
Squash the language code blocks and fix list.md (#865)
2023-10-17 01:06:00 +08:00
```src
[file]{build_tree}-[class]{}-[func]{build_tree}
```
Release Rust code to documents. (#656)
2023-07-26 11:00:53 +08:00
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
2023-12-02 06:21:34 +08:00
下图展示了构建二叉树的递归过程，各个节点是在向下“递”的过程中建立的，而各条边（引用）是在向上“归”的过程中建立的。
feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
=== "<1>"
Fix naming of the section build_binary_tree_problem
2023-07-17 02:18:53 +08:00
![构建二叉树的递归过程](build_binary_tree_problem.assets/built_tree_step1.png)
feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
=== "<2>"
Fix naming of the section build_binary_tree_problem
2023-07-17 02:18:53 +08:00
![built_tree_step2](build_binary_tree_problem.assets/built_tree_step2.png)
feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
=== "<3>"
Fix naming of the section build_binary_tree_problem
2023-07-17 02:18:53 +08:00
![built_tree_step3](build_binary_tree_problem.assets/built_tree_step3.png)
feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
=== "<4>"
Fix naming of the section build_binary_tree_problem
2023-07-17 02:18:53 +08:00
![built_tree_step4](build_binary_tree_problem.assets/built_tree_step4.png)
feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
=== "<5>"
Fix naming of the section build_binary_tree_problem
2023-07-17 02:18:53 +08:00
![built_tree_step5](build_binary_tree_problem.assets/built_tree_step5.png)
feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
=== "<6>"
Fix naming of the section build_binary_tree_problem
2023-07-17 02:18:53 +08:00
![built_tree_step6](build_binary_tree_problem.assets/built_tree_step6.png)
feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
=== "<7>"
Fix naming of the section build_binary_tree_problem
2023-07-17 02:18:53 +08:00
![built_tree_step7](build_binary_tree_problem.assets/built_tree_step7.png)
feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
=== "<8>"
Fix naming of the section build_binary_tree_problem
2023-07-17 02:18:53 +08:00
![built_tree_step8](build_binary_tree_problem.assets/built_tree_step8.png)
feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
=== "<9>"
Fix naming of the section build_binary_tree_problem
2023-07-17 02:18:53 +08:00
![built_tree_step9](build_binary_tree_problem.assets/built_tree_step9.png)
feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
Fix some figures.
2023-08-30 02:52:50 +08:00
每个递归函数内的前序遍历 `preorder` 和中序遍历 `inorder` 的划分结果如下图所示。
![每个递归函数中的划分结果](build_binary_tree_problem.assets/built_tree_overall.png)
feat: Add the chapter of divide and conquer (#609) * Add the chapter of divide and conquer. Add the section of divide and conquer algorithm. Add the section of build tree problem. * Update build_tree.py
2023-07-16 04:24:04 +08:00
设树的节点数量为 $n$ ，初始化每一个节点（执行一个递归函数 `dfs()` ）使用 $O(1)$ 时间。**因此总体时间复杂度为 $O(n)$** 。
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
2023-12-02 06:21:34 +08:00
哈希表存储 `inorder` 元素到索引的映射，空间复杂度为 $O(n)$ 。在最差情况下，即二叉树退化为链表时，递归深度达到 $n$ ，使用 $O(n)$ 的栈帧空间。**因此总体空间复杂度为 $O(n)$** 。
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