B树和B+树

简介

在计算机科学中，B树（英语：B-tree）是一种自平衡的树，能够保持数据有序。这种数据结构能够让查找数据、顺序访问、插入数据及删除的动作，都在对数时间内完成。B树，概括来说是一个一般化的二叉查找树（binary search tree），可以拥有多于2个子节点。与自平衡二叉查找树不同，B树为系统大块数据的读写操作做了优化。B树减少定位记录时所经历的中间过程，从而加快存取速度。B树这种数据结构可以用来描述外部存储。这种数据结构常被应用在数据库和文件系统的实现上。

定义

B-Tree is a self-balanced search tree with multiple keys in every node and more than two children for every node.

B树是一种自平衡的搜索树,每一个节点node都有多个keys，并且每个节点有2个子节点或者多于2个子节点。

A B+ tree is an N-ary tree with a variable but often large number of children per node. A B+ tree consists of a root, internal nodes and leaves.The root may be either a leaf or a node with two or more children

B+树是一个n叉排序树，通常每个节点有多个孩子，一棵B+树包含一个根节点、多个内部节点和叶子节点。根节点可能是一个叶子节点，也可能是一个包含两个或两个以上孩子节点的节点。

特征

B-Tree of Order m has the following properties...

• Property #1 - All the leaf nodes must be at same level.
• Property #2 - All nodes except root must have at least [m/2]-1 keys and maximum of m-1 keys.
• Property #3 - All non leaf nodes except root (i.e. all internal nodes) must have at least m/2 children.
• Property #4 - If the root node is a non leaf node, then it must have at least 2 children.
• Property #5 - A non leaf node with n-1 keys must have n number of children.
• Property #6 - All the key values within a node must be in Ascending Order.

搜寻方法

In a B-Ttree, the search operation is similar to that of Binary Search Tree. In a Binary search tree, the search process starts from the root node and every time we make a 2-way decision (we go to either left subtree or right subtree). In B-Tree also search process starts from the root node but every time we make n-way decision where n is the total number of children that node has. In a B-Ttree, the search operation is performed with O(log n) time complexity. The search operation is performed as follows...

• Step 1: Read the search element from the user
• Step 2: Compare, the search element with first key value of root node in the tree.
• Step 3: If both are matching, then display "Given node found!!!" and terminate the function
• Step 4: If both are not matching, then check whether search element is smaller or larger than that key value.
• Step 5: If search element is smaller, then continue the search process in left subtree.
• Step 6: If search element is larger, then compare with next key value in the same node and repeate step 3, 4, 5 and 6 until we found exact match or comparision completed with last key value in a leaf node.
• Step 7: If we completed with last key value in a leaf node, then display "Element is not found" and terminate the function.

大致的意思就是查询的条件A,和root节点值比较，如果相等则success,不相等则判断是否小于该节点，小于则查询左子树，大于则查询此时节点的下一个值，以此类推，直到此时节点最后一个值还是小于A,则查询右子树，直到找到或者结束查找。

插入方法

Insertion Operation in B-Tree
In a B-Tree, the new element must be added only at leaf node. That means, always the new keyValue is attached to leaf node only. The insertion operation is performed as follows...

• Step 1: Check whether tree is Empty.
• Step 2: If tree is Empty, then create a new node with new key value and insert into the tree as a root node.
• Step 3: If tree is Not Empty, then find a leaf node to which the new key value cab be added using Binary Search Tree logic.
• Step 4: If that leaf node has an empty position, then add the new key value to that leaf node by maintaining ascending order of key value within the node.
• Step 5: If that leaf node is already full, then split that leaf node by sending middle value to its parent node. Repeat tha same until sending value is fixed into a node.
• Step 6: If the spilting is occuring to the root node, then the middle value becomes new root node for the tree and the height of the tree is increased by one.

删除方法

参考8的链接,有图有真相，这里就摘录一些重点文字条件吧。

k：删除的值
x: k所在的节点
x.n: k所在节点的长度
t: k所在节点的层级

• If k is in the node x which is a leaf and x.n>=t,

Here you can straightaway delete k from x.

• If k is in the node x which is a leaf and x.n == (t-1)

  Find the immediate sibling y of x, the extreme key m of y, the parent p of x and the parent key l of k
  If y.n >= t:
      Move l into x
      Move m into p
      Delete k from x

  Find the immediate sibling y of x, the extreme key m of y, the parent p of x and the parent key l of k
  If y.n == t-1:
      Merge x and y
      Move down l to the new node as the median key
      Delete k from the new node

• If k is in the node x and x is an internal node (not a leaf)

  Find the child node y that precedes k (the node which is on the left side of k)
  If y.n >= t:
      Find the key k’ in y which is the predecessor of k
      Delete k’ recursively. (Here k’ can be another internal node as well. So we have to delete it recursively in the same way)
      Replace k with k’ in x

  Find the child node y that precedes k
  If y.n < t (or y.n == (t-1)):
      Find the child node z that follows k (the node which is on the right side of k)
      If z.n >= t:
          Find k’’ in z which is the successor of k
          Delete k’’ recursively
          Replace k with k’’ in x

  Find the child node y that precedes k and the child node z that follows k
  If y.n == (t-1) AND z.n == (t-1):
      Merge k and z to y
      Free memory of node z
      Recursively delete k from y

B-树和B+树区别

B和B+树的区别在于，B+树的非叶子结点只包含key信息，不包含data，每个节点的指针上限为2t而不是2t+1.所有的叶子结点和相连的节点使用链表相连，便于区间查找和遍历。

综述

磁盘存储和mysql的索引这一块用的比较多，以空间换时间来提升查找速度。（图片基本是从参考链接那边拿过来的，站在前人的肩膀上。）

参考

1. https://zh.wikipedia.org/wiki/B%E6%A0%91
2. http://blog.codinglabs.org/articles/theory-of-mysql-index.html
3. http://btechsmartclass.com/DS/U5_T3.html
4. http://www.cnblogs.com/yangecnu/p/Introduce-B-Tree-and-B-Plus-Tree.html
5. https://www.geeksforgeeks.org/b-tree-set-1-introduction-2/
6. https://www.geeksforgeeks.org/b-tree-set-1-insert-2/
7. https://www.geeksforgeeks.org/b-tree-set-3delete/
8. https://medium.com/@vijinimallawaarachchi/all-you-need-to-know-about-deleting-keys-from-b-trees-9090f3334b5c