Recursion. Presentation Subtitle. Brad Miller David Ranum 1 12/19/2005. Luther College. Binary Search Trees. 1 Department of Computer Science

Transcription

1 Presentation Subtitle Brad Miller David Ranum 1 1 Department of Computer Science Luther College 12/19/2005

2 Outline 1 Binary Search Trees

3 Outline Binary Search Trees 1 Binary Search Trees

4 BinaryTree() Create a new, empty binary tree. put(key,val) Add a new key-value pair to the tree. get(key) Given a key, return the value stored in the tree or None otherwise. delete_key(key) Delete the key-value pair from the tree. length() Return the number of key-value pairs stored in the tree. has_key(key) Return True if the given key is in the dictionary. operators We can use the above methods to overload the [] operators for both assignment and lookup. In addition, we can use has_key to override the in operator.

5 Outline Binary Search Trees 1 Binary Search Trees

6 A Simple Binary Search Tree

7 The Binary Search Tree Outer Class I 1 c l a s s BinarySearchTree: 2 def init (self): 3 self.root = None 4 self.size = d e f put(self,key,val): 7 i f self.root: 8 self.root.put(key,val) 9 e l s e : 10 self.root = TreeNode(key,val) 11 self.size = self.size def setitem (self,k,v): 14 self.put(k,v) def get(self,key): 17 i f self.root:

8 The Binary Search Tree Outer Class II 18 return self.root.get(key) 19 e l s e : 20 return None def getitem (self,key): 23 return self.get(key) d e f has_key(self,key): 26 i f self.root.get(key): 27 return True 28 e l s e : 29 return False def length(self): 32 return self.size def len (self):

9 The Binary Search Tree Outer Class III 35 return self.size d e f delete_key(self,key): 38 i f self.size > 1: 39 self.root.delete_key(key) 40 self.size = self.size-1 41 e l i f self.root.key == key: 42 self.root = None 43 self.size = self.size e l s e : 45 p r i n t error, bad key

10 Constructor for a TreeNode 1 d e f init (self,key,val,parent=none, 2 left=none,right=none): 3 self.key = key 4 self.payload = val 5 self.leftchild = left 6 self.rightchild = right 7 self.parent = parent

11 Inserting a new node Starting at the root of the tree, search the binary tree comparing the new key to the key in the current node. If the new key is less than the current node, search the left subtree. If the new key is greater than the current node, search the right subtree. When there is no left (or right) child to search, we have found the position in the tree where the new node should be installed. To add a node to the tree, create a new BinarySearchTree object and insert the object at the point discovered in the previous step.

12 Insert a New Node in a Binary Search Tree 1 d e f put(self,key,val): 2 i f key < self.key: 3 i f self.leftchild: 4 self.leftchild.put(key,val) 5 e l s e : 6 self.leftchild = TreeNode(key,val,self) 7 e l s e : 8 i f self.rightchild: 9 self.rightchild.put(key,val) 10 e l s e : 11 self.rightchild = TreeNode(key,val,self)

13 Overloading setitem 1 def setitem (self,k,v): 2 self.put(k,v)

14 Inserting a Node with Key =

15 Find the Value Stored with a Key 1 def get(self,key): 2 i f key == self.key: 3 return self.payload 4 e l i f key < self.key: 5 i f self.leftchild: 6 return self.leftchild.get(key) 7 e l s e : 8 return None 9 e l i f key > self.key: 10 i f self.rightchild: 11 r e t u r n self.rightchild.get(key) 12 e l s e : 13 return None 14 e l s e : 15 p r i n t error: this line should never be executed def getitem (self,key): 18 return self.get(key)

16 A Simple length Method I 1 2 def length(self): 3 return self.size 4 5 def len (self): 6 return self.size

17 1 The node to be deleted has no children (see Figure 13). 2 The node to be deleted has only one child (see Figure 15). 3 The node to be deleted has two children (see Figure 16).

18 Case 1: Deleting a Node with No Children 1 2 i f not (self.leftchild or self.rightchild): 3 p r i n t "removing a node with no childrend" 4 i f self == self.parent.leftchild: 5 self.parent.leftchild = None 6 e l s e : 7 self.parent.rightchild = None

19 Deleting Node 16, a Node Without Children

20 Case 2: Deleting a Node with One Child 1 e l i f (self.leftchild or self.rightchild) \ 2 and ( not (self.leftchild and self.rightchild)): 3 p r i n t "removing a node with one child" 4 i f self.leftchild: 5 i f self == self.parent.leftchild: 6 self.parent.leftchild = self.leftchild 7 e l s e : 8 self.parent.rightchild = self.leftchild 9 e l s e : 10 i f self == self.parent.leftchild: 11 self.parent.leftchild = self.rightchild 12 e l s e : 13 self.parent.rightchild = self.rightchild

21 Deleting Node 29, a Node That Has a Single Child

22 Deleting Node 5, a Node with Two Children successor

23 Case 3: Delete a Node with Two Children 1 e l s e : 2 succ = self.findsuccessor() 3 succ.spliceout() 4 i f self == self.parent.leftchild: 5 self.parent.leftchild = succ 6 e l s e : 7 self.parent.rightchild = succ 8 succ.leftchild = self.leftchild 9 succ.rightchild = self.rightchild

24 1 If the node has a right child, then the successor is the smallest key in the right subtree. 2 If the node has no right child and is the immediate left child of its parent, then the parent is the successor. 3 If the node is the immediate right child of its parent, and itself has no right child, then the successor to this node is the successor of its parent, excluding this node.

25 Finding the Successor 1 d e f findsuccessor(self): 2 succ = None 3 i f self.rightchild: 4 succ = self.rightchild.findmin() 5 e l s e : 6 i f self.parent.leftchild == self: 7 succ = self.parent 8 e l s e : 9 self.parent.rightchild = None 10 succ = self.parent.findsuccessor() 11 self.parent.rightchild = self 12 return succ

26 Finding the minimum child 1 def findmin(self): 2 n = self 3 while n.leftchild: 4 n = n.leftchild 5 p r i n t found min, key =, n.key 6 return n

27 Helper Method to Splice Out a Node 1 def spliceout(self): 2 i f ( not self.leftchild and not self.rightchild): 3 i f self == self.parent.leftchild: 4 self.parent.leftchild = None 5 e l s e : 6 self.parent.rightchild = None 7 e l i f (self.leftchild or self.rightchild): 8 i f self.leftchild: 9 i f self == self.parent.leftchild: 10 self.parent.leftchild = self.leftchild 11 e l s e : 12 self.parent.rightchild = self.leftchild 13 e l s e : 14 i f self == self.parent.leftchild: 15 self.parent.leftchild = self.rightchild 16 e l s e : 17 self.parent.rightchild = self.rightchild

28 Code for Deleting a Key I 1 d e f delete_key(self,key): 2 i f self.key == key: # do the removal 3 i f not (self.leftchild or self.rightchild): 4 i f self == self.parent.leftchild: 5 self.parent.leftchild = None 6 e l s e : 7 self.parent.rightchild = None 8 e l i f (self.leftchild or self.rightchild) and \ 9 ( not (self.leftchild and self.rightchild)): 10 i f self.leftchild: 11 i f self == self.parent.leftchild: 12 self.parent.leftchild = self.leftchild 13 e l s e : 14 self.parent.rightchild = self.leftchild 15 e l s e : 16 i f self == self.parent.leftchild: 17 self.parent.leftchild = self.rightchild

29 Code for Deleting a Key II 18 e l s e : 19 self.parent.rightchild = self.rightchild 20 e l s e : # replace self with successor 21 succ = self.findsuccessor() 22 succ.spliceout() 23 i f self == self.parent.leftchild: 24 self.parent.leftchild = succ 25 e l s e : 26 self.parent.rightchild = succ 27 succ.leftchild = self.leftchild 28 succ.rightchild = self.rightchild 29 e l s e : # continue looking 30 i f key < self.key: 31 i f self.leftchild: 32 self.leftchild.delete_key(key) 33 e l s e : 34 e l s e :

30 Code for Deleting a Key III 35 i f self.rightchild: 36 self.rightchild.delete_key(key) 37 e l s e : 38 p r i n t "trying to remove a non-existant node"

31 An Iterator for a Binary Search Tree 1 def iter (self): 2 i f self: 3 i f self.leftchild: 4 f o r elem i n self.leftchild: 5 yield elem 6 yield self.key 7 i f self.rightchild: 8 f o r elem i n self.rightchild: 9 yield elem

32 Outline Binary Search Trees 1 Binary Search Trees

33 A Skewed Binary Search Tree