B树及2-3树的python实现

B树(或称B-树)是一种适用于外查找的树，它是一种平衡的多叉树。

阶为M的B树具有下列结构特征：

1.树的根或者是一片树叶，或者其儿子数在2和M之间。

2.除根节点外的所有非树叶节点儿子数在┌M/2┐和 M之间。

3.所有的树叶都在相同的高度。

4.节点中包括n个关键字，n+1个指针，一般形式为： （n,P0,K1,P1,K2,P2,…,Kn,Pn）。每个结点中关键字从小到大排列，并且当该结点的孩子是非叶子结点时，该k-1个关键字正好是k个儿子包含的关键字的值域的分划。

 

对于任意一颗包含n个关键字的M阶B树，高度h满足：

h≤log┌m/2┐((N+1)/2 )+1

当B树的分支因子很大时，可以大大降低树的高度，B树的查找效率非常之高。

搜索B树

搜索B树与搜索二叉查找树的操作很类似，只是在每个节点所做的不是个二叉分支决定，而是根据该节点的子女数所做的多路分支决定。

向B树插入关键字

 1.向未满的节点插入关键字

2.向已满的节点添加关键字，需要将节点分裂为两个节点：

分裂一个节点有三种情况：

A：父节点未满

有两种情况，分裂leftchild与分裂middlechild：

B：父节点已满，需要将父节点分裂

有三种情况：

最后，特殊情况，产生新的根：

代码：

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class Node(object):     def __init__(self,key):         self.key1=key         self.key2=None         self.left=None         self.middle=None         self.right=None     def isLeaf(self):         return self.left is None and self.middle is None and self.right is None     def isFull(self):         return self.key2 is not None     def hasKey(self,key):         if (self.key1==key) or (self.key2 is not None and self.key2==key):             return True         else:             return False     def getChild(self,key):         if key<self.key1:             return self.left         elif self.key2 is None:             return self.middle         elif key<self.key2:             return self.middle         else:             return self.right class 2_3_Tree(object):     def __init__(self):         self.root=None     def get(self,key):         if self.root is None:             return None         else:             return self._get(self.root,key)     def _get(self,node,key):         if node is None:             return None         elif node.hasKey(key):             return node         else:             child=node.getChild(key)             return self._get(child,key)     def put(self,key):         if self.root is None:             self.root=Node(key)         else:             pKey,pRef=self._put(self.root,key)             if pKey is not None:                 newnode=Node(pKey)                 newnode.left=self.root                 newnode.middle=pRef                 self.root=newnode     def _put(self,node,key):         if node.hasKey(key):             return None,None         elif node.isLeaf():             return self._addtoNode(node,key,None)         else:             child=node.getChild(key)             pKey,pRef=self._put(child,key)             if pKey is None:                 return None,None             else:                 return self._addtoNode(node,pKey,pRef)                             def _addtoNode(self,node,key,pRef):         if node.isFull():             return self._splitNode(node,key,pRef)         else:             if key<node.key1:                 node.key2=node.key1                 node.key1=key                 if pRef is not None:                     node.right=node.middle                     node.middle=pRef             else:                 node.key2=key                 if pRef is not None:                     node.right=Pref             return None,None     def _splitNode(self,node,key,pRef):         newnode=Node(None)         if key<node.key1:             pKey=node.key1             node.key1=key             newnode.key1=node.key2             if pRef is not None:                 newnode.left=node.middle                 newnode.middle=node.right                 node.middle=pRef         elif key<node.key2:             pKey=key             newnode.key1=node.key2             if pRef is not None:                 newnode.left=Pref                 newnode.middle=node.right         else:             pKey=node.key2             newnode.key1=key             if pRef is not None:                 newnode.left=node.right                 newnode.middle=pRef         node.key2=None         return pKey,newnode