对于二叉查找树的每个节点Node，它的左子树中所有的关键字都小于Node的关键字，而右子树中的所有关键字都大于Node的关键字。
二叉查找树的平均深度是O(log N)。
1.初始化
class BinarySearchTree(object):
def __init__(self,key):
self.key=key
self.left=None
self.right=None
2.Find
def find(self,x):
if x==self.key:
return self
elif x<self.key and self.left:
return self.left.find(x)
elif x>self.key and self.right:
return self.right.find(x)
else:
return None
3.FindMin和FindMax
分别返回树中的最小元素与最大元素的位置。FindMin，从根开始并且只要有左儿子就向左进行查找，终止点是最小元素。FindMax则向右进行。
def findMin(self):
if self.left:
return self.left.findMin()
else:
return self
def findMax(self):
tree=self
if tree:
while tree.right:
tree=tree.right
return tree
4.Insert
为了将x插入到树Tree中，先用find查找，如果找到x，则什么也不做。否则，将x插入到遍历路径的最后一点。
来自《Problem Solving with Algorithms and Data Structures》的图片：
def insert(self,x):
if x<self.key:
if self.left:
self.left.insert(x)
else:
tree=BinarySearchTree(x)
self.left=tree
elif x>self.key:
if self.right:
self.right.insert(x)
else:
tree=BinarySearchTree(x)
self.right=tree
5.Delete
删除某节点有3种情况：
5.1 如果节点是一片树叶，那么可以立即被删除。
来自《Problem Solving with Algorithms and Data Structures》的图片：
5.2 如果节点只有一个儿子，则将此节点parent的指针指向此节点的儿子，然后删除。
来自《Problem Solving with Algorithms and Data Structures》的图片：
5.3 如果节点有两个儿子，则将其右子树的最小数据代替此节点的数据，并将其右子树的最小数据（不可能有左儿子，只有一个右儿子）删除。
来自《Problem Solving with Algorithms and Data Structures》的图片：
def delete(self,x):
if self.find(x):
if x<self.key:
self.left=self.left.delete(x)
return self
elif x>self.key:
self.right=self.right.delete(x)
return self
elif self.left and self.right:
key=self.right.findMin().key
self.key=key
self.right=self.right.delete(key)
return self
else:
if self.left:
return self.left
else:
return self.right
else:
return self
全部代码
class BinarySearchTree(object):
def __init__(self,key):
self.key=key
self.left=None
self.right=None
def find(self,x):
if x==self.key:
return self
elif x<self.key and self.left:
return self.left.find(x)
elif x>self.key and self.right:
return self.right.find(x)
else:
return None
def findMin(self):
if self.left:
return self.left.findMin()
else:
return self
def findMax(self):
tree=self
if tree:
while tree.right:
tree=tree.right
return tree
def insert(self,x):
if x<self.key:
if self.left:
self.left.insert(x)
else:
tree=BinarySearchTree(x)
self.left=tree
elif x>self.key:
if self.right:
self.right.insert(x)
else:
tree=BinarySearchTree(x)
self.right=tree
def delete(self,x):
if self.find(x):
if x<self.key:
self.left=self.left.delete(x)
return self
elif x>self.key:
self.right=self.right.delete(x)
return self
elif self.left and self.right:
key=self.right.findMin().key
self.key=key
self.right=self.right.delete(key)
return self
else:
if self.left:
return self.left
else:
return self.right
else:
return self
上述写法的缺点是很难处理空树的情况。
另一种类似于链表的写法
class TreeNode(object):
def __init__(self,key,left=None,right=None,parent=None):
self.key=key
self.left=left
self.right=right
self.parent=parent
def hasLeftChild(self):
return self.left
def hasRightChild(self):
return self.right
def isLeftChild(self):
return self.parent and self.parent.left==self
def isRightChild(self):
return self.parent and self.parent.right==self
class BSTree(object):
def __init__(self):
self.root=None
self.size=0
def length(self):
return self.size
def insert(self,x):
node=TreeNode(x)
if not self.root:
self.root=node
self.size+=1
else:
currentNode=self.root
while True:
if x<currentNode.key:
if currentNode.left:
currentNode=currentNode.left
else:
currentNode.left=node
node.parent=currentNode
self.size+=1
break
elif x>currentNode.key:
if currentNode.right:
currentNode=currentNode.right
else:
currentNode.right=node
node.parent=currentNode
self.size+=1
break
else:
break def find(self,key):
if self.root:
res=self._find(key,self.root)
if res:
return res
else:
return None
else:
return None
def _find(self,key,node):
if not node:
return None
elif node.key==key:
return node
elif key<node.key:
return self._find(key,node.left)
else:
return self._find(key,node.right)
def findMin(self):
if self.root:
current=self.root
while current.left:
current=current.left
return current
else:
return None
def _findMin(self,node):
if node:
current=node
while current.left:
current=current.left
return current
def findMax(self):
if self.root:
current=self.root
while current.right:
current=current.right
return current
else:
return None
def delete(self,key):
if self.size>1:
nodeToRemove=self.find(key)
if nodeToRemove:
self.remove(nodeToRemove)
self.size-=1
else:
raise KeyError,'Error, key not in tree'
elif self.size==1 and self.root.key==key:
self.root=None
self.size-=1
else:
raise KeyError('Error, key not in tree')
def remove(self,node):
if not node.left and not node.right: #node为树叶
if node==node.parent.left:
node.parent.left=None
else:
node.parent.right=None elif node.left and node.right: #有两个儿子
minNode=self._findMin(node.right)
node.key=minNode.key
self.remove(minNode) else: #有一个儿子
if node.hasLeftChild():
if node.isLeftChild():
node.left.parent=node.parent
node.parent.left=node.left
elif node.isRightChild():
node.left.parent=node.parent
node.parent.right=node.left
else: #node为根
self.root=node.left
node.left.parent=None
node.left=None
else:
if node.isLeftChild():
node.right.parent=node.parent
node.parent.left=node.right
elif node.isRightChild():
node.right.parent=node.parent
node.parent.right=node.right
else: #node为根
self.root=node.right
node.right.parent=None
node.right=None