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* 8个值得了解的树形数据结构

8个值得了解的树形数据结构

* Update 8-Useful-Tree-Data-Structures-Worth-Knowing.md

* Update 8-useful-tree-data-structures-worth-knowing.md

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> * 原文作者：[Vijini Mallawaarachchi](https://medium.com/@vijinimallawaarachchi)
> * 译文出自：[掘金翻译计划](https://github.com/xitu/gold-miner)
> * 本文永久链接：[https://github.com/xitu/gold-miner/blob/master/TODO1/8-useful-tree-data-structures-worth-knowing.md](https://github.com/xitu/gold-miner/blob/master/TODO1/8-useful-tree-data-structures-worth-knowing.md)
> * 译者：
> * 校对者：
> * 译者：[Amberlin1970](https://github.com/Amberlin1970)
> * 校对者：[PingHGao](https://github.com/PingHGao), [Jiangzhiqi4551](https://github.com/Jiangzhiqi4551)
# 8 Useful Tree Data Structures Worth Knowing
# 8 个值得了解的树形数据结构

What comes to your mind when you think of a tree? Roots, branches and leaves? A big oak tree with roots, branches and leaves may come to your mind. Similarly, in computer science, the tree data structure has roots, branches and leaves, but it is drawn upside-down. A tree is a hierarchical data structure which can represent relationships between different nodes. In this article, I will briefly introduce you to 8 types of tree data structures.
对于一棵树你会想到什么呢？树根、分支和树叶？一个有树根、分支和树叶的大橡树可能会在你脑海浮现。相似地，在计算机科学中，树形数据结构有根部、分支和树叶，但它是自上而下的。树是一种层级数据结构，以表达不同节点之间的关系。在这篇文章中，我将会简要地给你介绍 8 种树形数据机构。

#### Properties of a Tree
#### 树的属性

* A tree can contain no nodes or it can contain one special node called the **root** with zero or more subtrees.
* Every edge of the tree is directly or indirectly originated from the root.
* Every child has only one parent, but one parent can have many children.
* 树可以没有节点也可以包含一个特殊的节点**根节点**，该根节点可以包含零或多棵子树。
* 树的每条边都直接或间接地源于根节点。
* 每个孩子节点只有一个父节点，但每个父节点可以有很多孩子节点。

![Fig 1. Terminology of trees](https://cdn-images-1.medium.com/max/2000/1*PWJiwTxRdQy8A_Y0hAv5Eg.png)
![Fig 1. 树的相关术语](https://cdn-images-1.medium.com/max/2000/1*PWJiwTxRdQy8A_Y0hAv5Eg.png)

In this article, I will be briefly explaining the following 10 tree data structures with their usage.
在这篇文章中，我将会简要地解释下面 8 种树形数据结构及它们的用途。

1. General tree
2. Binary tree
3. Binary search tree
4. AVL tree
5. Red-black tree
6. Splay tree
7. Treap
8. B-tree
1. 普通树
2. 二叉树
3. 二叉搜索树
4. AVL树
5. 红黑树
6. 伸展树
7. 树堆
8. B树

## 1. General Tree
## 1. 普通树

A **general tree** is a tree data structure where there are no constraints on the hierarchical structure.
**普通树**是一种在层级结构上无限制的树形数据结构。

#### Properties
#### 属性

1. Follow properties of a tree.
2. A node can have any number of children.
1. 遵循树的属性。
2. 一个节点可以有任意数量的孩子。

![Fig 2. General tree](https://cdn-images-1.medium.com/max/2000/1*rInucvqb9X8bqM5yE143SQ.png)
![Fig 2. 普通树](https://cdn-images-1.medium.com/max/2000/1*rInucvqb9X8bqM5yE143SQ.png)

#### Usage
#### 用途

1. Used to store hierarchical data such as folder structures.
1. 用于存储如文件结构类的层级数据。

## 2. Binary Tree
## 2. 二叉树

A **binary tree** is a tree data structure where the following properties can be found.
**二叉树**是一种有如下属性的树形数据结构。

#### Properties
#### 属性

1. Follow properties of a tree.
2. A node can have at most two child nodes (children).
3. These two child nodes are known as the **left child** and **right child**.
1. 遵循树的属性。
2. 一个节点至多有两个孩子。
3. 这两个孩子分别称为**左孩子**和**右孩子**。

![Fig 3. Binary tree](https://cdn-images-1.medium.com/max/2000/1*abunFFnReygaqVt93xNr2A.png)
![Fig 3. 二叉树](https://cdn-images-1.medium.com/max/2000/1*abunFFnReygaqVt93xNr2A.png)

#### Usage
#### 用途

1. Used by compilers to build syntax trees.
2. Used to implement expression parsers and expression solvers.
3. Used to store router-tables in routers.
1. 在编译器中用于构造语法树。
2. 用于执行表达式解析和表达式求解。
3. 路由器中用于存储路由器表。

## 3. Binary Search Tree
## 3. 二叉搜索树

A **binary search tree** is a more constricted extension of a binary tree.
**二叉搜索树**是二叉树的一种更受限的扩展。

#### Properties
#### 属性

1. Follow properties of a binary tree.
2. Has a unique property known as the **binary-search-tree property**. This property states that the value (or key) of the left child of a given node should be less than or equal to the parent value and the value of the right child should be greater than or equal to the parent value.
1. 遵循二叉树的属性。
2. 有一个独特的属性，称作**二叉搜索树属性**。这个属性要求一个给定节点的左孩子的值（或键）应该小于或等于父节点的值，而右孩子的值应该大于或等于父节点的值。

![Fig 4. Binary search tree](https://cdn-images-1.medium.com/max/2000/1*jBgV9A847f_pHMbO67tcgw.png)
![Fig 4. 二叉搜索树](https://cdn-images-1.medium.com/max/2000/1*jBgV9A847f_pHMbO67tcgw.png)

#### Usage
#### 用途

1. Used to implement simple sorting algorithms.
2. Can be used as priority queues.
3. Used in many search applications where data are constantly entering and leaving.
1. 用于执行简单的排序算法。
2. 可以用作优先队列。
3. 适用于数据持续进出的很多搜索应用中。

## 4. AVL tree
## 4. AVL树

An **AVL tree** is a self-balancing binary search tree. This is the first tree introduced which automatically balances its height.
**平衡二叉树**是一种自我平衡的二叉搜索树。这是介绍的第一种自动平衡高度的树。

#### Properties
#### 属性

1. Follow properties of binary search trees.
2. Self-balancing.
3. Each node stores a value called a **balance factor** which is the difference in height between its left subtree and right subtree.
4. All the nodes must have a balance factor of -1, 0 or 1.
1. 遵循二叉搜索树的属性。
2. 自平衡。
3. 每一个节点储存了一个值，称为一个**平衡因子**，即为其左子树和右子树的高度差。
4. 所有的节点都必须有一个平衡因子且只能是 -1、0 或 1。

After performing insertions or deletions, if there is at least one node that does not have a balance factor of -1, 0 or 1 then rotations should be performed to balance the tree (self-balancing). You can read more about the rotation operations in my previous article from [**here**](https://towardsdatascience.com/self-balancing-binary-search-trees-101-fc4f51199e1d).
在进行插入或是删除后，如果有一个节点的平衡因子不是 -1、0 或 1，就必须通过旋转来平衡树（自平衡）。你可以在我前面的文章 [**这里**](https://towardsdatascience.com/self-balancing-binary-search-trees-101-fc4f51199e1d) 阅读到更多的旋转操作。

![Fig 5. AVL tree](https://cdn-images-1.medium.com/max/2000/1*aI575o1BBE3B4cAFUG73pw.png)
![Fig 5. AVL树](https://cdn-images-1.medium.com/max/2000/1*aI575o1BBE3B4cAFUG73pw.png)

#### Usage
#### 用途

1. Used in situations where frequent insertions are involved.
2. Used in Memory management subsystem of the Linux kernel to search memory regions of processes during preemption.
1. 用于需要频繁插入的情况。
2. 在 Linux 内核的内存管理子系统中用于在抢占期间搜索进程的内存区域。

## 5. Red-black tree
## 5. 红黑树

A red-black tree is a self-balancing binary search tree, where each node has a colour; red or black. The colours of the nodes are used to make sure that the tree remains approximately balanced during insertions and deletions.
红黑树是一种自平衡的二叉搜索树，每一个节点都有一种颜色：红或黑。节点的颜色用于确保树在插入和删除时保持大致的平衡。

#### Properties
#### 属性

1. Follow properties of binary search trees.
2. Self-balancing.
3. Each node is either red or black.
4. The root is black (sometimes omitted).
5. All leaves (denoted as NIL) are black.
6. If a node is red, then both its children are black.
7. Every path from a given node to any of its leaf nodes must go through the same number of black nodes.
1. 遵循二叉搜索树的属性。
2. 自平衡。
3. 每个节点或是红色或是黑色。
4. 根节点是黑色（有时省略）。
5. 所有叶子（标注为 NIL）是黑色。
6. 如果一个节点是红色，那它的孩子都是黑色。
7. 从一个给定的节点到其任意的叶子节点的每条路径都必须经过相同数目的黑色节点。

![Fig 6. AVL tree](https://cdn-images-1.medium.com/max/2000/1*11zvjUozpAenuez03oUeYA.png)
![Fig 6. 红黑树](https://cdn-images-1.medium.com/max/2000/1*11zvjUozpAenuez03oUeYA.png)

#### Usage
#### 用途

1. As a base for data structures used in computational geometry.
2. Used in the **Completely Fair Scheduler** used in current Linux kernels.
3. Used in the **epoll** system call implementation of Linux kernel.
1. 在计算几何中作为数据结构的基础。
2. 用于现在的 Linux 内核的**完全公平调度算法**中。
3. 用于 Linux 内核的 **epoll** 系统中。

## 6. Splay tree
## 6. 伸展树

A **splay tree** is a self-balancing binary search tree.
**伸展树**是一种自平衡的二叉搜索树。

#### Properties
#### 属性

1. Follow properties of binary search trees.
2. Self-balancing.
3. Recently accessed elements are quick to access again.
1. 遵循二叉搜索树的属性。
2. 自平衡。
3. 近期获取过的元素再次获取时速度更快。

After performing a search, insertion or deletion, splay trees perform an action called **splaying** where the tree is rearranged (using rotations) so that the particular element is placed at the root of the tree.
在搜索、插入或是删除后，伸展树会执行一个动作，称为**伸展**，在执行伸展动作时，树会被重新排列（使用旋转）将特定元素放置在树的根节点。

![Fig 7. Splay tree search](https://cdn-images-1.medium.com/max/2000/1*w5MA0XAEk1vX1lef4cUbdA.png)
![Fig 7. 伸展树搜索](https://cdn-images-1.medium.com/max/2000/1*w5MA0XAEk1vX1lef4cUbdA.png)

#### Usage
#### 用途

1. Used to implement caches
2. Used in garbage collectors.
3. Used in data compression
1. 用于实现缓存。
2. 用在垃圾收集器中。
3. 用于数据压缩。

## 7. Treap
## 7. 树堆

A **treap** (the name derived from **tree + heap**) is a binary search tree.
**树堆**（名字来源于**树**和**堆**的结合）是一种二叉搜索树。

#### Properties
#### 属性

1. Each node has two values; a **key** and a **priority**.
2. The keys follow the binary-search-tree property.
3. The priorities (which are random values) follow the heap property.
1. 每个节点有两个值：一个**键**和一个**优先级**。
2. 键遵循二叉搜索树的属性。
3. 优先级（随机值）遵循堆的属性。

![Fig 8. Treap (red coloured alphabetic keys follow BST property and blue coloured numeric values follow max heap order)](https://cdn-images-1.medium.com/max/2000/1*iH-zgLTHTHYe2E56aa2MWw.png)
![Fig 8. 树堆（红色的字母键遵循 BST 属性而蓝色的数字键遵循最大堆顺序）](https://cdn-images-1.medium.com/max/2000/1*iH-zgLTHTHYe2E56aa2MWw.png)

#### Usage
#### 用途

1. Used to maintain authorization certificates in public-key cryptosystems.
2. Can be used to perform fast set operations.
1. 用于维护公钥密码系统中的授权证书。
2. 可以用于执行快速的集合运算。

## 8. B-tree
## 8. B树

B tree is a self-balancing search tree and contains multiple nodes which keep data in sorted order. Each node has 2 or more children and consists of multiple keys.
B树是一种自平衡的搜索树，而且包含了多个排过序的节点。每个节点有 2 个或多个孩子且包含多个键。

#### Properties
#### 属性

1. Every node x has the following:
* x.n (the number of keys)
* x.keyᵢ (the keys stored in ascending order)
* x.leaf (whether x is a leaf or not)
2. Every node x has (x.n + 1) children.
3. The keys x.keyᵢ separate the ranges of keys stored in each sub-tree.
4. All the leaves have the same depth, which is the tree height.
5. Nodes have lower and upper bounds on the number of keys that can be stored. Here we consider a value t≥2, called **minimum degree** (or **branching factor**) of the B tree.
* The root must have at least one key.
* Every other node must have at least (t-1) keys and at most (2t-1) keys. Hence, every node will have at least t children and at most 2t children. We say the node is **full** if it has (2t-1) keys.
1. 每个节点 x 有如下（属性）：
* x.n（键的数量）
* x.keyᵢ（键以升序存储）
* x.leaf（x 是否是一个叶子）
2. 每个节点 x 有（x.n + 1）孩子。
3. 键 x.keyᵢ 分割了存储在每个子树中键的范围。
4. 所有的叶子有相等的深度，即为树的高度。
5. 节点有存储的键的数量的上界和下界。这里我们考虑一个值 t≥2，称为 B树的**最小度**（或**分支因子**）。
* 根节点必须至少有一个键。
* 每个其余节点必须有至少（t - 1）个键和至多（2t - 1）个键。因此，每个节点将会有至少 t 个孩子和至多 2t 个孩子。如果一个节点有（2t - 1）个键，我们称这个点是**满的**。

![Fig 9. B-tree](https://cdn-images-1.medium.com/max/2788/1*GXwr5PFqDNOOk8ae-8W5zA.png)
![Fig 9. B树](https://cdn-images-1.medium.com/max/2788/1*GXwr5PFqDNOOk8ae-8W5zA.png)

#### Usage
#### 用途

1. Used in database indexing to speed up the search.
2. Used in file systems to implement directories.
1. 用于数据库索引以加速搜索。
2. 在文件系统中用于实现目录。

## Final Thoughts
## 最后的思考

A cheat sheet for the time complexities of the data structure operations can be found in this [link](https://www.bigocheatsheet.com/).
数据结构操作的时间复杂度的备忘录可以在这个[链接](https://www.bigocheatsheet.com/)找到。

I hope you found this article useful as a simple introduction to tree structures. I would love to hear your thoughts. 😇
我希望这篇文章作为一个对树形结构的简单介绍对你是有用的。我很乐意听你的想法。😇

Thanks a lot for reading!
非常感谢阅读！

Cheers! 😃
祝愉快! 😃

## References
## 参考

- [1] Introduction to Algorithms (Third Edition) by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Livest and Clifford Stein.
- [1] 算法导论（第三版），作者：Thomas H.Cormen / Charles E.Leiserson / Ronald L.Rivest / Clifford Stein.
- [2] [https://en.wikipedia.org/wiki/List_of_data_structures](https://en.wikipedia.org/wiki/List_of_data_structures)

> 如果发现译文存在错误或其他需要改进的地方，欢迎到 [掘金翻译计划](https://github.com/xitu/gold-miner) 对译文进行修改并 PR，也可获得相应奖励积分。文章开头的 **本文永久链接** 即为本文在 GitHub 上的 MarkDown 链接。
> 如果发现译文存在错误或其他需要改进的地方，欢迎到 [掘金翻译计划](https://github.com/xitu/gold-miner) 对译文进行修改并 PR，也可获得相应奖励积分。文章开头的
*本文永久链接** 即为本文在 GitHub 上的 MarkDown 链接。

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