A binary search tree is an ordered binary tree. The left subtree contains nodes whose keys are less than the node’s key value, while the right subtree contains nodes whose keys are greater than or equal to the node’s key value.
Sometime, we add a reference to the parent node so that it will be more efficient for some operations such as find inorder successor etc. With the help of parent pointer, the action starts from the current node instead of root node. The time complexity reduces from O(h) to O(1). But for other operations, such as insert or delete, there will be more work. we have to add or remove reference to the parent for each node.
Table of Content
Define classes
As its name says, the TreeNode in binary search tree with parent add one more attribute to the regular binary tree node, parent. So there are four attributes, data, left, right and parent. data is the data stored in this node. left is the reference to the left child. right is the reference to right child. parent is reference to parent node.
After we define TreeNode, we can define BinarySearchTree class. It has one attribute, root. It is the node from where most tree operations start.
Please note, since binary search tree is ordered, in Java, BinarySearchTree’s generic data type extends Comparable interface, so that we can compare data in insert, delete or search operations.
Java
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public class BinarySearchTreeWithParent<T extends Comparable<T>> { //TreeNode with a parent reference static class TreeNode<T> { T data; TreeNode<T> left = null; TreeNode<T> right = null; TreeNode<T> parent = null; //Override, Time O(1), Space O(1) @Override public String toString() { return String.valueOf(data); } } TreeNode<T> root = null; } |
JavaScript
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class TreeNode { //Constructor, with parent pointer, Time O(1), Space O(1) TreeNode() { this.left = null; this.right = null; this.parent = null; } //Override, Time O(1), Space O(1) toString() { return this.data; } } class BinarySearchTreeWithParent { //Constructor, Time O(1), Space O(1) constructor() { this.root = null; } } |
Python
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class TreeNode : #Constructor, TreeNode with a parent reference, Time O(1) Space O(1) def __init__(self) : self.left = None self.right = None self.parent = None; #Override, Time O(1), Space O(1) def __str__(self): return str(self.data) class BinarySearchTreeWithParent : #Constructor, Time O(1) Space O(1) def __init__(self) : self.root = None |
Insert
In order to insert a value to a binary search tree, we have to find the right spot. From the root, we compare the input value with each node’s data. If the value is larger than the node’s data, we move to its right child. Otherwise move to its left child, Meanwhile we keep track the previous visited node and save in variable parent.
When we reach the node whose left or right child is null, we can add the new node with the input value as left or right child. Meantime, add a reference to the parent for this new node.
Java
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//Insert with parent pointer, Iteration, Time O(h) Space O(1), h is height of tree public void insert(T value) { TreeNode<T> newNode = new TreeNode<>(); newNode.data = value; if(root == null) { root = newNode; return; } TreeNode<T> curr = root; TreeNode<T> parent; while(true) { parent = curr; if(value.compareTo(curr.data) < 0) { curr = curr.left; if(curr == null) { parent.left = newNode; newNode.parent = parent; return; } } else { curr = curr.right; if(curr == null) { parent.right = newNode; newNode.parent =parent; return; } } } } |
JavaScript
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//Insert with parent pointer, Iteration, Time O(h) Space O(1), h is height of tree insert(value) { var newNode = new TreeNode(); newNode.data = value; if(this.root == null) { this.root = newNode; return; } var curr = this.root; var parent; while(true) { parent = curr; if(value < curr.data) { curr = curr.left; if(curr == null) { parent.left = newNode; newNode.parent =parent; return; } } else { curr = curr.right; if(curr == null) { parent.right = newNode; newNode.parent =parent; return; } } } } |
Python
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#Insert with parent pointer, Iteration, Time O(h) Space O(1), h is height of tree def insert(self, value) : newNode = TreeNode() newNode.data = value if self.root == None : self.root = newNode return curr = self.root parent = None while True : parent = curr if value < curr.data: curr = curr.left if curr == None: parent.left = newNode newNode.parent = parent return else : curr = curr.right if curr == None: parent.right = newNode newNode.parent =parent return |
Doodle
Delete by key
Delete a node in binary search tree with parent is similar to delete a node in binary search tree. First is to find the node with the data matches with the key. Then find its successor. In additional, we have to remember the parent of the deleted node, and point the successor’s parent to the old parent.
Java
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//Delete with parent pointer, Iteration, Time O(h) Space O(1) public TreeNode<T> delete(T key) { if (root == null ) return null; if (root.data == key){ //delete root root = deleteNode(root); //reset the root root.parent = null; return root; } TreeNode<T> curr = root; while (true) { if (key.compareTo(curr.data) > 0) { if (curr.right == null ) break; else if (curr.right.data == key) { TreeNode<T> oldParent = curr.right.parent; curr.right = deleteNode(curr.right); if (curr.right != null) curr.right.parent = oldParent; break; } curr = curr.right; } else { if (curr.left == null) break; else if (curr.left.data == key) { TreeNode<T> oldParent = curr.right.parent; curr.left = deleteNode(curr.left); if (curr.left != null) curr.left.parent = oldParent; break; } curr = curr.left; } } return root; } //Delete node, Time O(h), Space O(1) private TreeNode<T> deleteNode(TreeNode<T> node) { if (node == null) return null; if (node.right == null) return node.left; TreeNode<T> curr = node.right; while (curr.left != null) //find the left-most node curr = curr.left; curr.left = node.left; // put the original left under left most, if (node.left!=null) node.left.parent = curr; // link the new far left part to old far left return node.right; } |
JavaScript
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//Delete with parent pointer, Iteration, Time O(h) Space O(1) delete(key) { if (this.root == null ) return null; if (this.root.data == key){ //delete root this.root = this.deleteNode(this.root); //reset the root this.root.parent = null; return this.root; } var curr = this.root; while (true) { if (key > curr.data) { if (curr.right == null ) break; else if (curr.right.data == key) { let oldParent = curr.right.parent; curr.right = this.deleteNode(curr.right); if (curr.right != null) curr.right.parent = oldParent; break; } curr = curr.right; } else { if (curr.left == null) break; else if (curr.left.data == key) { let oldParent = curr.right.parent; curr.left = this.deleteNode(curr.left); if (curr.left != null) curr.left.parent = oldParent; break; } curr = curr.left; } } return this.root; } //Delete node, Time O(h), Space O(1) deleteNode(node) { if (node == null) return null; if (node.right == null) return node.left; var curr = node.right; while (curr.left != null) //find the left-most node curr = curr.left; curr.left = node.left; // put the original left under left most, if (node.left != null) node.left.parent = curr; // link the new far left part to old far left return node.right; } |
Python
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#Delete with parent pointer, Iteration, Time O(h) Space O(1) def delete(self, key) : if self.root == None : return None if self.root.data == key: #delete root self.root = self.deleteNode(self.root) #reset the root self.root.parent = None return self.root curr = self.root while True : if key > curr.data : if curr.right == None : break elif curr.right.data == key: oldParent = curr.right.parent curr.right = self.deleteNode(curr.right) if curr.right != None: curr.right.parent = oldParent break curr = curr.right else : if curr.left == None: break elif curr.left.data == key : oldParent = curr.right.parent curr.left = self.deleteNode(curr.left) if curr.left != None: curr.left.parent = oldParent break curr = curr.left return self.root #Delete node, Time O(h), Space O(1) def deleteNode(self, node) : if node == None: return None if node.right == None: return node.left curr = node.right while curr.left != None: #find the left-most node curr = curr.left curr.left = node.left # put the original left under left most, if node.left != None: node.left.parent = curr # link the new far left part to old far left return node.right |
Search by key
Search in binary search tree with parent is the same as search in binary search tree. There are two ways to search, Depth first search and Breadth first search.
Depth first search (DFS) starts from the root, and explores the child nodes as far as possible before backtracking. DFS is usually implemented with recursion or stack.
Breath First Search (BFS) starts from the root, and explores all its children before going to the next level. BFS is usually implemented with Queue.
Please refer to search in binary search tree for implementation.
Traversal
The traversal is to visit all nodes in the tree. It is the same as binary tree, including inorder, preorder, postorder and level order.
Inorder is to traverse the left subtree, visit the root, traverse the right subtree. In binary search tree, the output of the key should be in ascending order.
Preorder is to visit the root, traverse the left subtree, traverse the right subtree.
Postorder is to traverse the left subtree, traverse the right subtree, visit the root.
Level order is to traverse the nodes level by level.
please refer to the traversal of binary tree for the implementation.
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