Multi balanced tree --B-Tree (B tree)

　　Keywords: M-order, B trees, split, merge

　　Preface:

　　　　Prior to explain in detail, there is little to emphasize again the next: B- tree, is the B-tree. Because the original English name of the B-tree is B-tree, and many people like the domestic B-tree translated B- tree, in fact, this is a very bad translation, it is easy to misunderstand. As people might think B- tree is a tree, while the B-tree is a kind of tree. And in fact, B-tree refers to the B-tree. (B <=> B- Tree Tree). Quoted from: https://www.cnblogs.com/yfzhou/p/10290006.html

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1, B tree defines:

　　For the B-tree, we generally described as M (M> 2) Order tree B (M Order herein refers to the maximum number of subtrees of all nodes in the tree). For the B-tree, it must meet the following properties:

　　 Nature :

　　(1) the nature of the node:

• • All leaf nodes are at the same level;
  • Each node consists of a number and composition of the recording pointer . Wherein each node record number 1 smaller than the number of pointers, the pointer recorded every other.

　　(2) the nature of the pointer:

• • Each pointer to a child node;
    
  • Each node has a pointer up to M;
  • Non-leaf nodes of the root least two branches;
  • Leaf node contains a minimum of two records and a null pointer, the pointers are leaf nodes NULL;
  • Non-root non-leaf node has at least ceil ( M / 2 ) branches; (ceil ( M / 2 )> =. 3) (Analysis: When this value is equal to 2, then it becomes a binary tree; When this value is equal to 1, becomes the list, so this value must be greater than or equal to 3)
  • For any node: ceil (M / 2) <= d <= m. (Can be derived from the point) (d: indicates the degree of each node)

　　(3) the nature of the record:

• Each recorded by the [key, Data] composition, while the size of each record by the key sit right non descending;
  
• For any node in a record, all key values ​​recorded in the child node pointer which points to the left are smaller than the current record, is greater than the right pointer of the current record;
• For any node: ceil (M / 2) -1 <= k_count <= m-1. (K_count: indicates the number of records for each node)

A B-tree as follows:

 

Pictures taken from: http://blog.codinglabs.org/articles/theory-of-mysql-index.html

2, the characteristics of B-tree summary:

(1) Each node stores a [key, data]

(2) any one keyword appears once and only in one node;

(3) it is possible to search for the end of the non-leaf node;

(4) due to the limitations M / 2, and when the insertion node, if the node is full, the node needs to be split into two nodes each accounted M / 2; and delete nodes need to be two less than M sibling / 2 combined; (insert, delete operation to be solved the issue) ;

(5) B trees, like red-black trees are self-balancing trees.

3, B tree search:

(1) Search Process :

① starts from the root node, the nodes of the keyword in order to proceed equivalence determination;

② If the hit is finished, the current record is the corresponding keyword you want to find;

③ If there is no hit, the process proceeds to the son node belongs to the scope of the query key, the operation ① is repeated until the pointer is null corresponding to the son, or has been a leaf node. At this point then the tree does not exist in this keyword.

(2) search algorithm pseudocode on B-Tree as follows:

(Cited Code: http://blog.codinglabs.org/articles/theory-of-mysql-index.html )

1. BTree_Search(node, key) {
2.     if(node == null) return null;
3.     foreach(node.key)
4.     {
5.         if(node.key[i] == key) return node.data[i];
6.         if(node.key[i] > key) return BTree_Search(point[i]->node);
7.     }
8.     return BTree_Search(point[i+1]->node);
9. }
10. data = BTree_Search(root, my_key);

4、B树的插入、删除：

前言： 修复方式之分裂、合并、转移：

由于插入、删除数据记录会破坏B-Tree的性质，因此在插入删除时，需要对树进行一个分裂、合并、转移等操作以保持B-Tree性质。

①分裂：拆分当前节点的中间节记录到上层，与上层的节点的记录进行合并操作；

②合并：将记录与上层节点的记录进行合并；

③转移：需要不断将当前记录与其的相邻下级子节点中的记录进行交换，直到当前记录处于叶子节点中。（如果操作的节点记录不在叶子节点，那么就需要进行转移操作）

 

（1）插入：B树的插入总是在叶子节点上

如果插入新的数据记录到某个节点之后，节点的记录个数大于等于M，那么需要进行分裂操作。

（ 2）删除：

①如果节点不在叶子节点上，那么需要把它替换到叶子节点上。然后对叶子节点执行删除操作；

②如果删除当前节点的某个记录之后，节点的记录个数小于ceil(M/2)，那么需要进行合并操作；

③如果合并之后，上级节点记录个数小于等于ceil(M/2)，那么此时首先判断当前节点的兄弟节点的记录个数是否在小于ceil(m/2)，如果不是，那么取兄弟节点的一个记录与上级节点进行合并。否则，只能从上级节点的上级节点拉取一个记录下来进行合并。

 

注意：

①插入、删除的示例详见：https://www.cnblogs.com/yfzhou/p/10290006.html具

②体情况具体分析。遇到连锁反应（eg：分裂之后有需要分裂），就继续处理呗。

5、B树的节点的存储：

　　虽然B-Tree中不同节点存放的key和指针可能数量不一致，但是每个节点的域和上限是一致的，所以在实现中B-Tree往往对每个节点申请同等大小的空间。

 

 

引用： 

https://www.cnblogs.com/yfzhou/p/10290006.html

https://blog.csdn.net/maihilton/article/details/79900770

http://blog.codinglabs.org/articles/theory-of-mysql-index.html