Red-black tree

Introduction of the tree
- Trees have features are:
(1) each node has zero or more child nodes

(2) no parent node is called the root node

(3) Each non-root node has only one parent node and

(4) In addition to the root, each sub-node can be divided into a plurality of disjoint sub-trees
- Noun understand:
  - Node: refers to an element tree;
  - Of node: refers to the number of nodes of the subtree has a binary tree is not greater than 2;
  - Leaves: node 0 degree, also referred to as terminal nodes;
  - Height: the height of the leaf node is 1, the maximum height of the root node;
  - Parent: if a node contains a child node, this node is called a parent node to its child nodes

Child nodes: the child node is a parent node of the next layer.
- level node: starting from the root, root for the first layer, the second child of the root layer, and so on
- sibling node: The node share a common parent of mutual called sibling
- degree
- Depth (Depth) is the maximum height of the tree or hierarchical tree node.

Binary Tree
- Binary tree each node has at most two sub-trees tree structure.
- It has five basic forms: the binary tree may be an empty set; root is left empty subtree or right subtree; or the left and right subtree are all empty.
Why do we need a balanced binary tree?

To prevent this from happening, we hope to have an algorithm, our unbalanced binary sort tree into a balanced binary tree sort. This allows us to sort of binary tree structure optimization.
Balance factor
- The height of the left subtree of the node - the height of the right child node of the tree, not only about poor sub-tree height.
Balanced binary tree rotation
- LL型
- LR type
- Type RR
  - A rotation adjustment
- RL type
  - The first rotation, RR type
  - Second rotation, RR type adjustment
- I.
  - Add a new node 3
  - According to LL type, once a right-handed adjustment
- Scenario 2
  - Adding nodes 7
  - According to LL type, once a right-handed adjustment
- Three cases
  - Adding node 13
  - It requires two rotations, the first left-handed rotation type Cheng LL
  - According to LL type, a right-handed
- Four cases
  - Adding node 14
  - LL first rotary drive type, the first rotation
  - Second, based on the LL type, adjust
- Other add nodes to the right, the mirror is adjusted based on several cases, the first type is adjusted to RR and RR based secondary adjusted to balance.
  - Examples of a
  - Add 17 nodes
  - RR based on type, can be a rotating balance
  - Example Two
  - RR first rotary drive type, making adjustments
  - Based on RR type adjustment
Red-black tree Introduction

红黑树（Red Black Tree）是一种自平衡二叉查找树，是在计算机科学中用到的一种数据结构。

它是在1972年由Rudolf Bayer发明的，当时被称为平衡二叉B树（symmetric binary B-trees）。后来，在1978年被 Leo J. Guibas 和 Robert Sedgewick 修改为如今的 “红黑树” 。

R-B Tree，全称是Red-Black Tree，又称为“红黑树”，它一种特殊的二叉查找树。红黑树的每个节点上都有存储位表示节点的颜色，可以是红(Red)或黑(Black)。
红黑树的特性
- 性质1. 节点是红色或黑色。
- 性质2. 根节点是黑色。
- 性质3 每个叶节点（NIL节点，空节点）是黑色的。
- 性质4 每个红色节点的两个子节点都是黑色。（从每个叶子到根的所有路径上不能有两个连续的红色节点）
- 性质5. 从任一节点到其每个叶子的所有路径都包含相同数目的黑色节点。
左旋和右旋
- 左旋
- 右旋：
插入操作：

在对红黑树进行插入操作时，我们一般总是插入红色的节点，因为这样可以在插入过程中尽量避免对树的调整。那么，我们插入一个节点后，可能会使原树的哪些性质改变列?

如果插入的节点是根节点，性质2会被破坏，如果插入节点的父节点是红色，则会破坏性质4。因此，总而言之，插入一个红色节点只会破坏性质2或性质4。

我们的恢复策略很简单，插入修复具体操作情况：
1. 情况1：插入的是根节点。
原树是空树，此情况只会违反性质2。

对策：直接把此节点涂为黑色。
1. 情况2：插入的节点的父节点是黑色。
  
  此不会违反性质2和性质4，红黑树没有被破坏。
  
  对策：什么也不做。
2. 情况3：当前节点的父节点是红色且祖父节点的另一个子节点（叔叔节点）是红色。
  
  此时父节点的父节点一定存在，否则插入前就已不是红黑树。与此同时，又分为父节点是祖父节点的左子还是右子，对于对称性，我们只要解开一个方向就可以了。在此，我们只考虑父节点为祖父左子的情况。同时，还可以分为当前节点是其父节点的左子还是右子，但是处理方式是一样的。我们将此归为同一类。
  
  Solution: The parent and uncle of the current node black, painted red grandparent, grandparent point to the current node, the current node restart from the new algorithm.
  
  3 for the previous case, a change [the current node is node 4]:
Changes:
1. Case 4: The parent of the current node is red, black uncle node, the current node is the right child of its parent
  
  Solution: the parent node of the current node as the new current node to the new current node as a fulcrum left.
  
  As shown below, the pre-change [the current node is node 7]:
Changes:
1. Case 5: The parent of the current node is red, black uncle node, the current node is the left child of its parent
  
  Solution: parent node to black, to red grandparent node, grandparent node as a fulcrum in a right-handed
  
  As shown below [the current node is node 2]
Changes:

to sum up:

After the above case of three operations to repair a schematic view of the insertion case 3, case 4, case 5 and the like, one will find that the repair of the latter case is inserted into a series of 4, 5 are for the case 3 is inserted into the case after the node 4, the operation performed, However, pointing to the current node pointer N has been changing. So, you can think of: the entire down, 3,4,5 situation is a complete insertion of the repair of operational processes.