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