Develop Reference

Recursive Add method for BST using Javascript not working

Below is the implementation of a BST with an insertion function for it. currently, the code wouldn't work; It would just spit out Tree { root: null }
When i tried to debug it, it seems that it successfully adds the new Node to the correct spot, but once it returns from the function, all that data is lost and it ends up not inserting anything.
here is the code:
class Node {
constructor(value) {
this.value = value
this.left = null;
this.right = null;
}
}
class Tree {
constructor() {
this.root = null
}
insert(value) {
const insertHelper = (value, node) => {
if (node === null) {
node = new Node(value)
return null
} else if (node.value === node.value) {
console.log("Value exists.")
return null;
} else if (node.value < node.value) {
return this.insertHelper(node, node.right)
} else {
return this.insertHelper(node, node.left)
}
}
return insertHelper(value, this.root)
}
}
var tree = new Tree;
tree.insert(10)
tree.insert(5)
console.log(tree);

Several issues:
this.root is never modified. Function arguments are passed by value, so if you pass this.root as argument, and the function assigns a new value to the corresponding parameter variable node, this will not affect this.root. The solution is to let the helper function return the new value of the node that is passed as argument, so you can assign it back to the root (or other node).
At several places you compare node.value with node.value. That is a mistake. The comparison should involve value.
The recursive calls pass node as first argument, while the function expects the value as first argument.
Here is the corrected code:
class Node {
constructor(value) {
this.value = value;
this.left = null;
this.right = null;
}
}
class Tree {
constructor() {
this.root = null;
}
insert(value) {
const insertHelper = (value, node) => {
if (node === null) {
node = new Node(value);
} else if (node.value === value) {
console.log("Value exists.");
} else if (node.value < value) {
node.right = insertHelper(value, node.right);
} else {
node.left = insertHelper(value, node.left);
}
return node;
}
this.root = insertHelper(value, this.root);
}
}
var tree = new Tree;
tree.insert(10);
tree.insert(5);
console.log(tree);
NB: use semi-colons explicitly. Relying on the automatic semi-colon insertion is asking for trouble. One day it will hit you.

Recursive functions to iterative. Binary tree post-order traverse [duplicate]

What is the algorithm for doing a post order traversal of a binary tree WITHOUT using recursion?

Here's the version with one stack and without a visited flag:
private void postorder(Node head) {
if (head == null) {
return;
}
LinkedList<Node> stack = new LinkedList<Node>();
stack.push(head);
while (!stack.isEmpty()) {
Node next = stack.peek();
boolean finishedSubtrees = (next.right == head || next.left == head);
boolean isLeaf = (next.left == null && next.right == null);
if (finishedSubtrees || isLeaf) {
stack.pop();
System.out.println(next.value);
head = next;
}
else {
if (next.right != null) {
stack.push(next.right);
}
if (next.left != null) {
stack.push(next.left);
}
}
}
}

Here's a link which provides two other solutions without using any visited flags.
https://leetcode.com/problems/binary-tree-postorder-traversal/
This is obviously a stack-based solution due to the lack of parent pointer in the tree. (We wouldn't need a stack if there's parent pointer).
We would push the root node to the stack first. While the stack is not empty, we keep pushing the left child of the node from top of stack. If the left child does not exist, we push its right child. If it's a leaf node, we process the node and pop it off the stack.
We also use a variable to keep track of a previously-traversed node. The purpose is to determine if the traversal is descending/ascending the tree, and we can also know if it ascend from the left/right.
If we ascend the tree from the left, we wouldn't want to push its left child again to the stack and should continue ascend down the tree if its right child exists. If we ascend the tree from the right, we should process it and pop it off the stack.
We would process the node and pop it off the stack in these 3 cases:
The node is a leaf node (no children)
We just traverse up the tree from the left and no right child exist.
We just traverse up the tree from the right.

Here's a sample from wikipedia:
nonRecursivePostorder(rootNode)
nodeStack.push(rootNode)
while (! nodeStack.empty())
currNode = nodeStack.peek()
if ((currNode.left != null) and (currNode.left.visited == false))
nodeStack.push(currNode.left)
else
if ((currNode.right != null) and (currNode.right.visited == false))
nodeStack.push(currNode.right)
else
print currNode.value
currNode.visited := true
nodeStack.pop()

This is the approach I use for iterative, post-order traversal. I like this approach because:
It only handles a single transition per loop-cycle, so it's easy to follow.
A similar solution works for in-order and pre-order traversals
Code:
enum State {LEFT, RIGHT, UP, CURR}
public void iterativePostOrder(Node root) {
Deque<Node> parents = new ArrayDeque<>();
Node curr = root;
State state = State.LEFT;
while(!(curr == root && state == State.UP)) {
switch(state) {
case LEFT:
if(curr.left != null) {
parents.push(curr);
curr = curr.left;
} else {
state = RIGHT;
}
break;
case RIGHT:
if(curr.right != null) {
parents.push(curr);
curr = curr.right;
state = LEFT;
} else {
state = CURR;
}
break;
case CURR:
System.out.println(curr);
state = UP;
break;
case UP:
Node child = curr;
curr = parents.pop();
state = child == curr.left ? RIGHT : CURR;
break;
default:
throw new IllegalStateException();
}
}
}
Explanation:
You can think about the steps like this:
Try LEFT
if left-node exists: Try LEFT again
if left-node does not exist: Try RIGHT
Try RIGHT
If a right node exists: Try LEFT from there
If no right exists, you're at a leaf: Try CURR
Try CURR
Print current node
All nodes below have been executed (post-order): Try UP
Try UP
If node is root, there is no UP, so EXIT
If coming up from left, Try RIGHT
If coming up from right, Try CURR

Here is a solution in C++ that does not require any storage for book keeping in the tree.
Instead it uses two stacks. One to help us traverse and another to store the nodes so we can do a post traversal of them.
std::stack<Node*> leftStack;
std::stack<Node*> rightStack;
Node* currentNode = m_root;
while( !leftStack.empty() || currentNode != NULL )
{
if( currentNode )
{
leftStack.push( currentNode );
currentNode = currentNode->m_left;
}
else
{
currentNode = leftStack.top();
leftStack.pop();
rightStack.push( currentNode );
currentNode = currentNode->m_right;
}
}
while( !rightStack.empty() )
{
currentNode = rightStack.top();
rightStack.pop();
std::cout << currentNode->m_value;
std::cout << "\n";
}

import java.util.Stack;
public class IterativePostOrderTraversal extends BinaryTree {
public static void iterativePostOrderTraversal(Node root){
Node cur = root;
Node pre = root;
Stack<Node> s = new Stack<Node>();
if(root!=null)
s.push(root);
System.out.println("sysout"+s.isEmpty());
while(!s.isEmpty()){
cur = s.peek();
if(cur==pre||cur==pre.left ||cur==pre.right){// we are traversing down the tree
if(cur.left!=null){
s.push(cur.left);
}
else if(cur.right!=null){
s.push(cur.right);
}
if(cur.left==null && cur.right==null){
System.out.println(s.pop().data);
}
}else if(pre==cur.left){// we are traversing up the tree from the left
if(cur.right!=null){
s.push(cur.right);
}else if(cur.right==null){
System.out.println(s.pop().data);
}
}else if(pre==cur.right){// we are traversing up the tree from the right
System.out.println(s.pop().data);
}
pre=cur;
}
}
public static void main(String args[]){
BinaryTree bt = new BinaryTree();
Node root = bt.generateTree();
iterativePostOrderTraversal(root);
}
}

// the java version with flag
public static <T> void printWithFlag(TreeNode<T> root){
if(null == root) return;
Stack<TreeNode<T>> stack = new Stack<TreeNode<T>>();
stack.add(root);
while(stack.size() > 0){
if(stack.peek().isVisit()){
System.out.print(stack.pop().getValue() + " ");
}else{
TreeNode<T> tempNode = stack.peek();
if(tempNode.getRight()!=null){
stack.add(tempNode.getRight());
}
if(tempNode.getLeft() != null){
stack.add(tempNode.getLeft());
}
tempNode.setVisit(true);
}
}
}

Depth first, post order, non recursive, without stack
When you have parent:
node_t
{
left,
right
parent
}
traverse(node_t rootNode)
{
bool backthreading = false
node_t node = rootNode
while(node <> 0)
if (node->left <> 0) and backthreading = false then
node = node->left
continue
endif
>>> process node here <<<
if node->right <> 0 then
lNode = node->right
backthreading = false
else
node = node->parent
backthreading = true
endif
endwhile

The logic of Post order traversal without using Recursion
In Postorder traversal, the processing order is left-right-current. So we need to visit the left section first before visiting other parts. We will try to traverse down to the tree as left as possible for each node of the tree. For each current node, if the right child is present then push it into the stack before pushing the current node while root is not NULL/None. Now pop a node from the stack and check whether the right child of that node exists or not. If it exists, then check whether it's same as the top element or not. If they are same then it indicates that we are not done with right part yet, so before processing the current node we have to process the right part and for that pop the top element(right child) and push the current node back into the stack. At each time our head is the popped element. If the current element is not the same as the top and head is not NULL then we are done with both the left and right section so now we can process the current node. We have to repeat the previous steps until the stack is empty.
def Postorder_iterative(head):
if head is None:
return None
sta=stack()
while True:
while head is not None:
if head.r:
sta.push(head.r)
sta.push(head)
head=head.l
if sta.top is -1:
break
head = sta.pop()
if head.r is not None and sta.top is not -1 and head.r is sta.A[sta.top]:
x=sta.pop()
sta.push(head)
head=x
else:
print(head.val,end = ' ')
head=None
print()

void postorder_stack(Node * root){
stack ms;
ms.top = -1;
if(root == NULL) return ;
Node * temp ;
push(&ms,root);
Node * prev = NULL;
while(!is_empty(ms)){
temp = peek(ms);
/* case 1. We are nmoving down the tree. */
if(prev == NULL || prev->left == temp || prev->right == temp){
if(temp->left)
push(&ms,temp->left);
else if(temp->right)
push(&ms,temp->right);
else {
/* If node is leaf node */
printf("%d ", temp->value);
pop(&ms);
}
}
/* case 2. We are moving up the tree from left child */
if(temp->left == prev){
if(temp->right)
push(&ms,temp->right);
else
printf("%d ", temp->value);
}
/* case 3. We are moving up the tree from right child */
if(temp->right == prev){
printf("%d ", temp->value);
pop(&ms);
}
prev = temp;
}
}

Please see this full Java implementation. Just copy the code and paste in your compiler. It will work fine.
import java.util.LinkedList;
import java.util.Queue;
import java.util.Stack;
class Node
{
Node left;
String data;
Node right;
Node(Node left, String data, Node right)
{
this.left = left;
this.right = right;
this.data = data;
}
public String getData()
{
return data;
}
}
class Tree
{
Node node;
//insert
public void insert(String data)
{
if(node == null)
node = new Node(null,data,null);
else
{
Queue<Node> q = new LinkedList<Node>();
q.add(node);
while(q.peek() != null)
{
Node temp = q.remove();
if(temp.left == null)
{
temp.left = new Node(null,data,null);
break;
}
else
{
q.add(temp.left);
}
if(temp.right == null)
{
temp.right = new Node(null,data,null);
break;
}
else
{
q.add(temp.right);
}
}
}
}
public void postorder(Node node)
{
if(node == null)
return;
postorder(node.left);
postorder(node.right);
System.out.print(node.getData()+" --> ");
}
public void iterative(Node node)
{
Stack<Node> s = new Stack<Node>();
while(true)
{
while(node != null)
{
s.push(node);
node = node.left;
}
if(s.peek().right == null)
{
node = s.pop();
System.out.print(node.getData()+" --> ");
if(node == s.peek().right)
{
System.out.print(s.peek().getData()+" --> ");
s.pop();
}
}
if(s.isEmpty())
break;
if(s.peek() != null)
{
node = s.peek().right;
}
else
{
node = null;
}
}
}
}
class Main
{
public static void main(String[] args)
{
Tree t = new Tree();
t.insert("A");
t.insert("B");
t.insert("C");
t.insert("D");
t.insert("E");
t.postorder(t.node);
System.out.println();
t.iterative(t.node);
System.out.println();
}
}

Here I am pasting different versions in c# (.net) for reference:
(for in-order iterative traversal you may refer to: Help me understand Inorder Traversal without using recursion)
wiki (http://en.wikipedia.org/wiki/Post-order%5Ftraversal#Implementations) (elegant)
Another version of single stack
(#1 and #2: basically uses the fact that in post order traversal the right child node is visited before visiting the actual node - so, we simply rely on the check that if stack top's right child is indeed the last post order traversal node thats been visited - i have added comments in below code snippets for details)
Using Two stacks version (ref: http://www.geeksforgeeks.org/iterative-postorder-traversal/)
(easier: basically post order traversal reverse is nothing but pre order traversal with a simple tweak that right node is visited first, and then left node)
Using visitor flag (easy)
Unit Tests
~
public string PostOrderIterative_WikiVersion()
{
List<int> nodes = new List<int>();
if (null != this._root)
{
BinaryTreeNode lastPostOrderTraversalNode = null;
BinaryTreeNode iterativeNode = this._root;
Stack<BinaryTreeNode> stack = new Stack<BinaryTreeNode>();
while ((stack.Count > 0)//stack is not empty
|| (iterativeNode != null))
{
if (iterativeNode != null)
{
stack.Push(iterativeNode);
iterativeNode = iterativeNode.Left;
}
else
{
var stackTop = stack.Peek();
if((stackTop.Right != null)
&& (stackTop.Right != lastPostOrderTraversalNode))
{
//i.e. last traversal node is not right element, so right sub tree is not
//yet, traversed. so we need to start iterating over right tree
//(note left tree is by default traversed by above case)
iterativeNode = stackTop.Right;
}
else
{
//if either the iterative node is child node (right and left are null)
//or, stackTop's right element is nothing but the last traversal node
//(i.e; the element can be popped as the right sub tree have been traversed)
var top = stack.Pop();
Debug.Assert(top == stackTop);
nodes.Add(top.Element);
lastPostOrderTraversalNode = top;
}
}
}
}
return this.ListToString(nodes);
}
Here is post order traversal with one stack (my version)
public string PostOrderIterative()
{
List<int> nodes = new List<int>();
if (null != this._root)
{
BinaryTreeNode lastPostOrderTraversalNode = null;
BinaryTreeNode iterativeNode = null;
Stack<BinaryTreeNode> stack = new Stack<BinaryTreeNode>();
stack.Push(this._root);
while(stack.Count > 0)
{
iterativeNode = stack.Pop();
if ((iterativeNode.Left == null)
&& (iterativeNode.Right == null))
{
nodes.Add(iterativeNode.Element);
lastPostOrderTraversalNode = iterativeNode;
//make sure the stack is not empty as we need to peek at the top
//for ex, a tree with just root node doesn't have to enter loop
//and also node Peek() will throw invalidoperationexception
//if it is performed if the stack is empty
//so, it handles both of them.
while(stack.Count > 0)
{
var stackTop = stack.Peek();
bool removeTop = false;
if ((stackTop.Right != null) &&
//i.e. last post order traversal node is nothing but right node of
//stacktop. so, all the elements in the right subtree have been visted
//So, we can pop the top element
(stackTop.Right == lastPostOrderTraversalNode))
{
//in other words, we can pop the top if whole right subtree is
//traversed. i.e. last traversal node should be the right node
//as the right node will be traverse once all the subtrees of
//right node has been traversed
removeTop = true;
}
else if(
//right subtree is null
(stackTop.Right == null)
&& (stackTop.Left != null)
//last traversal node is nothing but the root of left sub tree node
&& (stackTop.Left == lastPostOrderTraversalNode))
{
//in other words, we can pop the top of stack if right subtree is null,
//and whole left subtree has been traversed
removeTop = true;
}
else
{
break;
}
if(removeTop)
{
var top = stack.Pop();
Debug.Assert(stackTop == top);
lastPostOrderTraversalNode = top;
nodes.Add(top.Element);
}
}
}
else
{
stack.Push(iterativeNode);
if(iterativeNode.Right != null)
{
stack.Push(iterativeNode.Right);
}
if(iterativeNode.Left != null)
{
stack.Push(iterativeNode.Left);
}
}
}
}
return this.ListToString(nodes);
}
Using two stacks
public string PostOrderIterative_TwoStacksVersion()
{
List<int> nodes = new List<int>();
if (null != this._root)
{
Stack<BinaryTreeNode> postOrderStack = new Stack<BinaryTreeNode>();
Stack<BinaryTreeNode> rightLeftPreOrderStack = new Stack<BinaryTreeNode>();
rightLeftPreOrderStack.Push(this._root);
while(rightLeftPreOrderStack.Count > 0)
{
var top = rightLeftPreOrderStack.Pop();
postOrderStack.Push(top);
if(top.Left != null)
{
rightLeftPreOrderStack.Push(top.Left);
}
if(top.Right != null)
{
rightLeftPreOrderStack.Push(top.Right);
}
}
while(postOrderStack.Count > 0)
{
var top = postOrderStack.Pop();
nodes.Add(top.Element);
}
}
return this.ListToString(nodes);
}
With visited flag in C# (.net):
public string PostOrderIterative()
{
List<int> nodes = new List<int>();
if (null != this._root)
{
BinaryTreeNode iterativeNode = null;
Stack<BinaryTreeNode> stack = new Stack<BinaryTreeNode>();
stack.Push(this._root);
while(stack.Count > 0)
{
iterativeNode = stack.Pop();
if(iterativeNode.visted)
{
//reset the flag, for further traversals
iterativeNode.visted = false;
nodes.Add(iterativeNode.Element);
}
else
{
iterativeNode.visted = true;
stack.Push(iterativeNode);
if(iterativeNode.Right != null)
{
stack.Push(iterativeNode.Right);
}
if(iterativeNode.Left != null)
{
stack.Push(iterativeNode.Left);
}
}
}
}
return this.ListToString(nodes);
}
The definitions:
class BinaryTreeNode
{
public int Element;
public BinaryTreeNode Left;
public BinaryTreeNode Right;
public bool visted;
}
string ListToString(List<int> list)
{
string s = string.Join(", ", list);
return s;
}
Unit Tests
[TestMethod]
public void PostOrderTests()
{
int[] a = { 13, 2, 18, 1, 5, 17, 20, 3, 6, 16, 21, 4, 14, 15, 25, 22, 24 };
BinarySearchTree bst = new BinarySearchTree();
foreach (int e in a)
{
string s1 = bst.PostOrderRecursive();
string s2 = bst.PostOrderIterativeWithVistedFlag();
string s3 = bst.PostOrderIterative();
string s4 = bst.PostOrderIterative_WikiVersion();
string s5 = bst.PostOrderIterative_TwoStacksVersion();
Assert.AreEqual(s1, s2);
Assert.AreEqual(s2, s3);
Assert.AreEqual(s3, s4);
Assert.AreEqual(s4, s5);
bst.Add(e);
bst.Delete(e);
bst.Add(e);
s1 = bst.PostOrderRecursive();
s2 = bst.PostOrderIterativeWithVistedFlag();
s3 = bst.PostOrderIterative();
s4 = bst.PostOrderIterative_WikiVersion();
s5 = bst.PostOrderIterative_TwoStacksVersion();
Assert.AreEqual(s1, s2);
Assert.AreEqual(s2, s3);
Assert.AreEqual(s3, s4);
Assert.AreEqual(s4, s5);
}
Debug.WriteLine(string.Format("PostOrderIterative: {0}", bst.PostOrderIterative()));
Debug.WriteLine(string.Format("PostOrderIterative_WikiVersion: {0}", bst.PostOrderIterative_WikiVersion()));
Debug.WriteLine(string.Format("PostOrderIterative_TwoStacksVersion: {0}", bst.PostOrderIterative_TwoStacksVersion()));
Debug.WriteLine(string.Format("PostOrderIterativeWithVistedFlag: {0}", bst.PostOrderIterativeWithVistedFlag()));
Debug.WriteLine(string.Format("PostOrderRecursive: {0}", bst.PostOrderRecursive()));
}

Python with 1 stack and no flag:
def postorderTraversal(self, root):
ret = []
if not root:
return ret
stack = [root]
current = None
while stack:
previous = current
current = stack.pop()
if previous and ((previous is current) or (previous is current.left) or (previous is current.right)):
ret.append(current.val)
else:
stack.append(current)
if current.right:
stack.append(current.right)
if current.left:
stack.append(current.left)
return ret
And what is better is with similar statements, in order traversal works too
def inorderTraversal(self, root):
ret = []
if not root:
return ret
stack = [root]
current = None
while stack:
previous = current
current = stack.pop()
if None == previous or previous.left is current or previous.right is current:
if current.right:
stack.append(current.right)
stack.append(current)
if current.left:
stack.append(current.left)
else:
ret.append(current.val)
return ret

I have not added the node class as its not particularly relevant or any test cases, leaving those as an excercise for the reader etc.
void postOrderTraversal(node* root)
{
if(root == NULL)
return;
stack<node*> st;
st.push(root);
//store most recent 'visited' node
node* prev=root;
while(st.size() > 0)
{
node* top = st.top();
if((top->left == NULL && top->right == NULL))
{
prev = top;
cerr<<top->val<<" ";
st.pop();
continue;
}
else
{
//we can check if we are going back up the tree if the current
//node has a left or right child that was previously outputted
if((top->left == prev) || (top->right== prev))
{
prev = top;
cerr<<top->val<<" ";
st.pop();
continue;
}
if(top->right != NULL)
st.push(top->right);
if(top->left != NULL)
st.push(top->left);
}
}
cerr<<endl;
}
running time O(n) - all nodes need to be visited
AND space O(n) - for the stack, worst case tree is a single line linked list

It's very nice to see so many spirited approaches to this problem. Quite inspiring indeed!
I came across this topic searching for a simple iterative solution for deleting all nodes in my binary tree implementation. I tried some of them, and I tried something similar found elsewhere on the Net, but none of them were really to my liking.
The thing is, I am developing a database indexing module for a very specific purpose (Bitcoin Blockchain indexing), and my data is stored on disk, not in RAM. I swap in pages as needed, doing my own memory management. It's slower, but fast enough for the purpose, and with having storage on disk instead of RAM, I have no religious bearings against wasting space (hard disks are cheap).
For that reason my nodes in my binary tree have parent pointers. That's (all) the extra space I'm talking about. I need the parents because I need to iterate both ascending and descending through the tree for various purposes.
Having that in my mind, I quickly wrote down a little piece of pseudo-code on how it could be done, that is, a post-order traversal deleting nodes on the fly. It's implemented and tested, and became a part of my solution. And it's pretty fast too.
The thing is: It gets really, REALLY, simple when nodes have parent pointers, and furthermore since I can null out the parent's link to the "just departed" node.
Here's the pseudo-code for iterative post-order deletion:
Node current = root;
while (current)
{
if (current.left) current = current.left; // Dive down left
else if (current.right) current = current.right; // Dive down right
else
{
// Node "current" is a leaf, i.e. no left or right child
Node parent = current.parent; // assuming root.parent == null
if (parent)
{
// Null out the parent's link to the just departing node
if (parent.left == current) parent.left = null;
else parent.right = null;
}
delete current;
current = parent;
}
}
root = null;
If you are interested in a more theoretical approach to coding complex collections (such as my binary tree, which is really a self-balancing red-black-tree), then check out these links:
http://opendatastructures.org/versions/edition-0.1e/ods-java/6_2_BinarySearchTree_Unbala.html
http://opendatastructures.org/versions/edition-0.1e/ods-java/9_2_RedBlackTree_Simulated_.html
https://www.cs.auckland.ac.nz/software/AlgAnim/red_black.html
Happy coding :-)
Søren Fog
http://iprotus.eu/

1.1 Create an empty stack
2.1 Do following while root is not NULL
a) Push root's right child and then root to stack.
b) Set root as root's left child.
2.2 Pop an item from stack and set it as root.
a) If the popped item has a right child and the right child
is at top of stack, then remove the right child from stack,
push the root back and set root as root's right child.
b) Else print root's data and set root as NULL.
2.3 Repeat steps 2.1 and 2.2 while stack is not empty.

Here is the Java implementation with two stacks
public static <T> List<T> iPostOrder(BinaryTreeNode<T> root) {
if (root == null) {
return Collections.emptyList();
}
List<T> result = new ArrayList<T>();
Deque<BinaryTreeNode<T>> firstLevel = new LinkedList<BinaryTreeNode<T>>();
Deque<BinaryTreeNode<T>> secondLevel = new LinkedList<BinaryTreeNode<T>>();
firstLevel.push(root);
while (!firstLevel.isEmpty()) {
BinaryTreeNode<T> node = firstLevel.pop();
secondLevel.push(node);
if (node.hasLeftChild()) {
firstLevel.push(node.getLeft());
}
if (node.hasRightChild()) {
firstLevel.push(node.getRight());
}
}
while (!secondLevel.isEmpty()) {
result.add(secondLevel.pop().getData());
}
return result;
}
Here is the unit tests
#Test
public void iterativePostOrderTest() {
BinaryTreeNode<Integer> bst = BinaryTreeUtil.<Integer>fromInAndPostOrder(new Integer[]{4,2,5,1,6,3,7}, new Integer[]{4,5,2,6,7,3,1});
assertThat(BinaryTreeUtil.iPostOrder(bst).toArray(new Integer[0]), equalTo(new Integer[]{4,5,2,6,7,3,1}));
}

/**
* This code will ensure holding of chain(links) of nodes from the root to till the level of the tree.
* The number of extra nodes in the memory (other than tree) is height of the tree.
* I haven't used java stack instead used this ParentChain.
* This parent chain is the link for any node from the top(root node) to till its immediate parent.
* This code will not require any altering of existing BinaryTree (NO flag/parent on all the nodes).
*
* while visiting the Node 11; ParentChain will be holding the nodes 9 -> 8 -> 7 -> 1 where (-> is parent)
*
* 1
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ \
2 7
/ \ /
/ \ /
/ \ /
/ \ /
3 6 8
/ \ /
/ \ /
4 5 9
/ \
10 11
*
* #author ksugumar
*
*/
public class InOrderTraversalIterative {
public static void main(String[] args) {
BTNode<String> rt;
String[] dataArray = {"1","2","3","4",null,null,"5",null,null,"6",null,null,"7","8","9","10",null,null,"11",null,null,null,null};
rt = BTNode.buildBTWithPreOrder(dataArray, new Counter(0));
BTDisplay.printTreeNode(rt);
inOrderTravesal(rt);
}
public static void postOrderTravesal(BTNode<String> root) {
ParentChain rootChain = new ParentChain(root);
rootChain.Parent = new ParentChain(null);
while (root != null) {
//Going back to parent
if(rootChain.leftVisited && rootChain.rightVisited) {
System.out.println(root.data); //Visit the node.
ParentChain parentChain = rootChain.Parent;
rootChain.Parent = null; //Avoid the leak
rootChain = parentChain;
root = rootChain.root;
continue;
}
//Traverse Left
if(!rootChain.leftVisited) {
rootChain.leftVisited = true;
if (root.left != null) {
ParentChain local = new ParentChain(root.left); //It is better to use pool to reuse the instances.
local.Parent = rootChain;
rootChain = local;
root = root.left;
continue;
}
}
//Traverse RIGHT
if(!rootChain.rightVisited) {
rootChain.rightVisited = true;
if (root.right != null) {
ParentChain local = new ParentChain(root.right); //It is better to use pool to reuse the instances.
local.Parent = rootChain;
rootChain = local;
root = root.right;
continue;
}
}
}
}
class ParentChain {
BTNode<String> root;
ParentChain Parent;
boolean leftVisited = false;
boolean rightVisited = false;
public ParentChain(BTNode<String> node) {
this.root = node;
}
#Override
public String toString() {
return root.toString();
}
}

void display_without_recursion(struct btree **b)
{
deque< struct btree* > dtree;
if(*b)
dtree.push_back(*b);
while(!dtree.empty() )
{
struct btree* t = dtree.front();
cout << t->nodedata << " " ;
dtree.pop_front();
if(t->right)
dtree.push_front(t->right);
if(t->left)
dtree.push_front(t->left);
}
cout << endl;
}

import java.util.Stack;
class Practice
{
public static void main(String arr[])
{
Practice prc = new Practice();
TreeNode node1 = (prc).new TreeNode(1);
TreeNode node2 = (prc).new TreeNode(2);
TreeNode node3 = (prc).new TreeNode(3);
TreeNode node4 = (prc).new TreeNode(4);
TreeNode node5 = (prc).new TreeNode(5);
TreeNode node6 = (prc).new TreeNode(6);
TreeNode node7 = (prc).new TreeNode(7);
node1.left = node2;
node1.right = node3;
node2.left = node4;
node2.right = node5;
node3.left = node6;
node3.right = node7;
postOrderIteratively(node1);
}
public static void postOrderIteratively(TreeNode root)
{
Stack<Entry> stack = new Stack<Entry>();
Practice prc = new Practice();
stack.push((prc).new Entry(root, false));
while (!stack.isEmpty())
{
Entry entry = stack.pop();
TreeNode node = entry.node;
if (entry.flag == false)
{
if (node.right == null && node.left == null)
{
System.out.println(node.data);
} else
{
stack.push((prc).new Entry(node, true));
if (node.right != null)
{
stack.push((prc).new Entry(node.right, false));
}
if (node.left != null)
{
stack.push((prc).new Entry(node.left, false));
}
}
} else
{
System.out.println(node.data);
}
}
}
class TreeNode
{
int data;
int leafCount;
TreeNode left;
TreeNode right;
public TreeNode(int data)
{
this.data = data;
}
public int getLeafCount()
{
return leafCount;
}
public void setLeafCount(int leafCount)
{
this.leafCount = leafCount;
}
public TreeNode getLeft()
{
return left;
}
public void setLeft(TreeNode left)
{
this.left = left;
}
public TreeNode getRight()
{
return right;
}
public void setRight(TreeNode right)
{
this.right = right;
}
#Override
public String toString()
{
return "" + this.data;
}
}
class Entry
{
Entry(TreeNode node, boolean flag)
{
this.node = node;
this.flag = flag;
}
TreeNode node;
boolean flag;
#Override
public String toString()
{
return node.toString();
}
}
}

I was looking for a code snippet that performs well and is simple to customise. Threaded trees are not “simple”. Double stack solution requires O(n) memory. LeetCode solution and solution by tcb have extra checks and pushes...
Here is one classic algorithm translated into C that worked for me:
void postorder_traversal(TreeNode *p, void (*visit)(TreeNode *))
{
TreeNode *stack[40]; // simple C stack, no overflow check
TreeNode **sp = stack;
TreeNode *last_visited = NULL;
for (; p != NULL; p = p->left)
*sp++ = p;
while (sp != stack) {
p = sp[-1];
if (p->right == NULL || p->right == last_visited) {
visit(p);
last_visited = p;
sp--;
} else {
for (p = p->right; p != NULL; p = p->left)
*sp++ = p;
}
}
}
IMHO this algorithm is easier to follow than well performing and readable wikipedia.org / Tree_traversal pseudocode. For glorious details see answers to binary tree exercises in Knuth’s Volume 1.

Here is a Python version too ::
class Node:
def __init__(self,data):
self.data = data
self.left = None
self.right = None
def postOrderIterative(root):
if root is None :
return
s1 = []
s2 = []
s1.append(root)
while(len(s1)>0):
node = s1.pop()
s2.append(node)
if(node.left!=None):
s1.append(node.left)
if(node.right!=None):
s1.append(node.right)
while(len(s2)>0):
node = s2.pop()
print(node.data)
root = Node(1)
root.left = Node(2)
root.right = Node(3)
root.left.left = Node(4)
root.left.right = Node(5)
root.right.left = Node(6)
root.right.right = Node(7)
postOrderIterative(root)
Here is the output ::

So you can use one stack to do a post order traversal.
private void PostOrderTraversal(Node pos) {
Stack<Node> stack = new Stack<Node>();
do {
if (pos==null && (pos=stack.peek().right)==null) {
for (visit(stack.peek()); stack.pop()==(stack.isEmpty()?null:stack.peek().right); visit(stack.peek())) {}
} else if(pos!=null) {
stack.push(pos);
pos=pos.left;
}
} while (!stack.isEmpty());
}

Two methods to perform Post Order Traversal without Recursion:
1. Using One HashSet of Visited Nodes and One stack for backtracking:
private void postOrderWithoutRecursion(TreeNode root) {
if (root == null || root.left == null && root.right == null) {
return;
}
Stack<TreeNode> stack = new Stack<>();
Set<TreeNode> visited = new HashSet<>();
while (!stack.empty() || root != null) {
if (root != null) {
stack.push(root);
visited.add(root);
root = root.left;
} else {
root = stack.peek();
if (root.right == null || visited.contains(root.right)) {
System.out.print(root.val+" ");
stack.pop();
root = null;
} else {
root = root.right;
}
}
}
}
Time Complexity: O(n)
Space Complexity: O(2n)
2. Using Tree Altering method:
private void postOrderWithoutRecursionAlteringTree(TreeNode root) {
if (root == null || root.left == null && root.right == null) {
return;
}
Stack<TreeNode> stack = new Stack<>();
while (!stack.empty() || root != null) {
if (root != null) {
stack.push(root);
root = root.left;
} else {
root = stack.peek();
if (root.right == null) {
System.out.print(root.val+" ");
stack.pop();
root = null;
} else {
TreeNode temp = root.right;
root.right = null;
root = temp;
}
}
}
}
Time Complexity: O(n)
Space Complexity: O(n)
TreeNode Class:
public class TreeNode {
public int val;
public TreeNode left;
public TreeNode right;
public TreeNode(int x) {
val = x;
}
}

Here's a short (the walker is 3 lines) version that I needed to write in Python for a general tree. Of course, works for a more limited binary tree too. Tree is a tuple of the node and list of children. It only has one stack. Sample usage shown.
def postorder(tree):
def do_something(x): # Your function here
print(x),
def walk_helper(root_node, calls_to_perform):
calls_to_perform.append(partial(do_something, root_node[0]))
for child in root_node[1]:
calls_to_perform.append(partial(walk_helper, child, calls_to_perform))
calls_to_perform = []
calls_to_perform.append(partial(walk_helper, tree, calls_to_perform))
while calls_to_perform:
calls_to_perform.pop()()
postorder(('a', [('b', [('c', []), ('d', [])])]))
d
c
b
a

The simplest solution, it may look like not the best answer but it is easy to understand. And I believe if you have understood the solution then you can modify it to make the best possible solution
// using two stacks
public List<Integer> postorderTraversal(TreeNode root){
Stack<TreeNode> st=new Stack<>();
Stack<TreeNode> st2=new Stack<>();
ArrayList<Integer> al = new ArrayList<Integer>();
if(root==null)
return al;
st.push(root); //push the root to 1st stack
while(!st.isEmpty())
{
TreeNode curr=st.pop();
st2.push(curr);
if(curr.left!=null)
st.push(curr.left);
if(curr.right!=null)
st.push(curr.right);
}
while(!st2.isEmpty())
al.add(st2.pop().val);
//this ArrayList contains the postorder traversal
return al;
}

Simple Intuitive solution in python.
while stack:
node = stack.pop()
if node:
if isinstance(node,TreeNode):
stack.append(node.val)
stack.append(node.right)
stack.append(node.left)
else:
post.append(node)
return post

This is what I've come up for post order iterator:
class PostOrderIterator
implements Iterator<T> {
private Stack<Node<T>> stack;
private Node<T> prev;
PostOrderIterator(Node<T> root) {
this.stack = new Stack<>();
recurse(root);
this.prev = this.stack.peek();
}
private void recurse(Node<T> node) {
if(node == null) {
return;
}
while(node != null) {
stack.push(node);
node = node.left;
}
recurse(stack.peek().right);
}
#Override
public boolean hasNext() {
return !stack.isEmpty();
}
#Override
public T next() {
if(stack.peek().right != this.prev) {
recurse(stack.peek().right);
}
Node<T> next = stack.pop();
this.prev = next;
return next.value;
}
}
Basically, the main idea is that you should think how the initialization process puts the first item to print on the top of the stack, while the rest of the stack follow the nodes that would have been touched by the recursion. The rest would just then become a lot easier to nail.
Also, from design perspective, PostOrderIterator is an internal class exposed via some factory method of the tree class as an Iterator<T>.

In post-order traversal, he left child of a node is visited first, followed by its right child, and finally the node itself.
This tree traversal method is similar to depth first search (DFS) traversal of a graph.
Time Complexity: O(n)
Space Complexity: O(n)
Below is the iterative implementation of post-order traversal in python:
class Node:
def __init__(self, data=None, left=None, right=None):
self.data = data
self.left = left
self.right = right
def post_order(node):
if node is None:
return []
stack = []
nodes = []
last_node_visited = None
while stack or node:
if node:
stack.append(node)
node = node.left
else:
peek_node = stack[-1]
if peek_node.right and last_node_visited != peek_node.right:
node = peek_node.right
else:
nodes.append(peek_node.data)
last_node_visited = stack.pop()
return nodes
def main():
'''
Construct the below binary tree:
15
/ \
/ \
/ \
10 20
/ \ / \
8 12 16 25
'''
root = Node(15)
root.left = Node(10)
root.right = Node(20)
root.left.left = Node(8)
root.left.right = Node(12)
root.right.left = Node(16)
root.right.right = Node(25)
print(post_order(root))
if __name__ == '__main__':
main()

For writing iterative equivalents of these recursive methods, we can first understand how the recursive methods themselves execute over the program's stack. Assuming that the nodes do not have their parent pointer, we need to manage our own "stack" for the iterative variants.
One way to start is to see the recursive method and mark the locations where a call would "resume" (fresh initial call, or after a recursive call returns). Below these are marked as "RP 0", "RP 1" etc ("Resume Point"). For the case of postorder traversal:
void post(node *x)
{
/* RP 0 */
if(x->lc) post(x->lc);
/* RP 1 */
if(x->rc) post(x->rc);
/* RP 2 */
process(x);
}
Its iterative variant:
void post_i(node *root)
{
node *stack[1000];
int top;
node *popped;
stack[0] = root;
top = 0;
popped = NULL;
#define POPPED_A_CHILD() \
(popped && (popped == curr->lc || popped == curr->rc))
while(top >= 0)
{
node *curr = stack[top];
if(!POPPED_A_CHILD())
{
/* type (x: 0) */
if(curr->rc || curr->lc)
{
if(curr->rc) stack[++top] = curr->rc;
if(curr->lc) stack[++top] = curr->lc;
popped = NULL;
continue;
}
}
/* type (x: 2) */
process(curr);
top--;
popped = curr;
}
}
The code comments with (x: 0) and (x: 2) correspond to the "RP 0" and "RP 2" resume points in the recursive method.
By pushing both the lc and rc pointers together, we have removed the need of keeping the post(x) invocation at resume-point 1 while the post(x->lc) completes its execution. That is, we could directly shift the node to "RP 2", bypassing "RP 1". So, there is no node kept on stack at "RP 1" stage.
The POPPED_A_CHILD macro helps us deduce one of the two resume-points ("RP 0", or "RP 2").

How to save last searched when using search on input?

I am trying to write an algorithm that saves the recently searched keywords. The issue is that the user is not triggering the search but instead the search is being triggered every time something is typed. Currently, I am trying to save the last searched keyword and checking if the next one starts with the last one in order to discard the last one.
async _persistSearch(term) {
const serachLog = await this._getLastSearchedFile()
console.log("[SEARCH]", serachLog);
if (Object.keys(serachLog) == 0) {
serachLog.searches = [term];
serachLog.lastUpdated = Date.now()
} else if (term.startsWith(this.lastSearhed)) {
const toPersist = serachLog.searches.filter((value) => value != this.lastSearhed).push(term);
serachLog.searches = toPersist;
} else {
serachLog.searches.push(term);
}
this.lastSearhed = term;
await writeJSONFile(this.searchesFile, serachLog);
}
For example for an input [so, som, some, somet, someth,somethi, somethin, something, somethin] to be [something]

You should use Trie(Prefix Tree) to save searched keywords. Say you are only dealing with lower-case English letters, then each node can have up to 26 children nodes. For each new search keyword, call the startsWith method to check if this new keyword is any prefix of the previous keywords.
Read about this https://en.wikipedia.org/wiki/Trie if you need more explanations about why a Trie should be used in your case.
A java implementation of this data structure is as following.
class Trie {
class TrieNode{
private char c;
private Map<Character, TrieNode> map;
private boolean isLeaf;
TrieNode() {
this.map = new HashMap<>();
}
TrieNode(char c) {
this.c = c;
this.map = new HashMap<>();
}
}
private TrieNode root;
/** Initialize your data structure here. */
public Trie() {
root = new TrieNode();
}
/** Inserts a word into the trie. */
public void insert(String word) {
TrieNode currNode = root;
for(int i = 0; i < word.length(); i++) {
char currChar = word.charAt(i);
if(!currNode.map.containsKey(currChar)) {
TrieNode newNode = new TrieNode(currChar);
currNode.map.put(currChar, newNode);
currNode = newNode;
}
else {
currNode = currNode.map.get(currChar);
}
if(i == word.length() - 1) {
currNode.isLeaf = true;
}
}
}
/** Returns if the word is in the trie. */
public boolean search(String word) {
TrieNode currNode = root;
for(int i = 0; i < word.length(); i++) {
if(!currNode.map.containsKey(word.charAt(i))) {
return false;
}
currNode = currNode.map.get(word.charAt(i));
if( i == word.length() - 1 && !currNode.isLeaf) {
return false;
}
}
return true;
}
/** Returns if there is any word in the trie that starts with the given prefix. */
public boolean startsWith(String prefix) {
TrieNode currNode = root;
for(int i = 0; i < prefix.length(); i++) {
if(!currNode.map.containsKey(prefix.charAt(i))) {
return false;
}
currNode = currNode.map.get(prefix.charAt(i));
}
return true;
}
}

Remove operation on binary search tree with only 1 node (the root)

Started writing the removal function for an unbalanced BST structure. Manually running some tests for the first case (node has no children). Decided to run it on a tree of size 1 (just the root), and for some reason it does not seem to be reassigning the root to null the way I'm expecting it to on line 3 of this statement:
return direction ?
parent[direction] :
node = null;
Then when I run inOrderTraversal on the single node tree, which should just console.log each node, and return undefined for a null tree (what I'm expecting) it simply prints the 55 as it does before the removal.
It seems to be working for all other cases where the node to remove has no children.
Here's the fiddle: https://jsfiddle.net/uvdrmwh0/6/
And the code:
"use strict";
function Node(value, left = null, right = null) {
return {
value,
left,
right
};
}
function insert(x, root) {
let currNode = root;
while (currNode) {
if (x < currNode.value) {
if (currNode.left) {
currNode = currNode.left;
} else {
currNode.left = Node(x);
return;
}
} else if (x > currNode.value) {
if (currNode.right) {
currNode = currNode.right;
} else {
currNode.right = Node(x);
return;
}
} else if (x === currNode.value) {
throw new Error("cannot insert node with the same value as an existing node");
} else {
throw new Error("undefined behavior in insert");
}
}
throw new Error("failed to insert");
}
function remove(x, node, parent = null, direction = null) {
if (node === null) return;
if (node.value === x) {
if (!node.left && !node.right) {
return direction ?
parent[direction] = null :
node = null;
} else if (node.left && !node.right) {
//TODO
}
//TODO
}
direction = x < node.value ? "left" : "right";
remove(x, node[direction], node, direction);
}
function inOrderTraversal(node) {
if (node === null) return;
inOrderTraversal(node.left);
console.log(node.value);
inOrderTraversal(node.right);
}
function BinarySearchTree(seed) {
if (!Array.isArray(seed)) {
throw new Error("BinarySearchTree must be seeded with an array");
}
let root = Node(seed[0]);
seed.slice(1).forEach(x => {
insert(x, root);
});
return root;
}
let bst = BinarySearchTree([55]);
inOrderTraversal(bst);
console.log("---------after removal---------");
remove(55, bst);
inOrderTraversal(bst);
Update:
I've noticed things like this work:
let x = { a: 1 };
function changeProperty(obj, key, newValue) {
obj[key] = newValue;
}
changeProperty(x, "a", "hello");
console.log(x.a); //prints hello
But this doesn't:
function reassignObject(obj) {
obj = { a: "some new value" };
}
reassignObject(x);
console.log(x.a); //still prints hello
It seems you can reassign properties of an object (pointers within an object) and it will change the outside reference, but reassigning the reference to the object itself doesn't?

The following change should make it work:
console.log("---------after removal---------");
bst = remove(55, bst); //change here
The changes to node happen locally in remove function. So you should set the bst to whatever is received back from remove function.
The important thing to understand here is how does javascript pass the arguments. I hope this helps.

How to search a node in a non ordered binary tree?

I'm building a non ordered binary tree that looks like this
1
/ \
30 2
\ / \
31 4 3
/ \
34 32
.... and so on
My implementation has been created as follows
function BinarySearchTree() {
this._root = null;
}
BinarySearchTree.prototype = {
//restore constructor
constructor: BinarySearchTree,
add: function(value,whereToAdd,nodeToAdd){
//create a new item object, place data in
var node = {
value: value,
left: null,
right: null
},
//used to traverse the structure
current;
//special case: no items in the tree yet
if (this._root === null || whereToAdd === null || whereToAdd === undefined){
this._root = node;
} else {
current = nodeToAdd;
while(true){
//if the new value is less than this node's value, go left
if (whereToAdd === "no"){
//if there's no left, then the new node belongs there
if (current.left === null){
current.left = node;
break;
} else {
current = current.left;
}
//if the new value is greater than this node's value, go right
} else if (whereToAdd === "yes"){
//if there's no right, then the new node belongs there
if (current.right === null){
current.right = node;
break;
} else {
current = current.right;
}
//if the new value is equal to the current one, just ignore
} else {
break;
}
}
}
},
traverse: function(process){
//helper function
function inOrder(node){
if (node){
//traverse the left subtree
if (node.left !== null){
inOrder(node.left);
}
//call the process method on this node
process.call(this, node);
//traverse the right subtree
if (node.right !== null){
inOrder(node.right);
}
}
}
//start with the root
inOrder(this._root);
},
contains: function(value){
var found = false,
current = this._root;
//make sure there's a node to search
while(!found && current){
//if the value is less than the current node's, go left
if (value < current.value){
current = current.left;
//if the value is greater than the current node's, go right
} else if (value > current.value){
current = current.right;
//values are equal, found it!
} else {
found = true;
}
}
//only proceed if the node was found
return current;
},
size: function(){
var length = 0;
this.traverse(function(node){
length++;
});
return length;
},
toArray: function(){
var result = [];
this.traverse(function(node){
result.push(node.value);
});
return result;
},
toString: function(){
return this.toArray().toString();
},
};
function ArmTheTree(Tree){
Tree.add(1,null,null);
Tree.add(30,"no",Tree.contains(1));
Tree.add(31,"yes",Tree.contains(30));
Tree.add(32,"yes",Tree.contains(31));
Tree.add(33,"no",Tree.contains(32));
Tree.add(34,"no",Tree.contains(31));
Tree.add(35,"yes",Tree.contains(34));
Tree.add(36,"yes",Tree.contains(35));
Tree.add(2,"yes",Tree.contains(1));
Tree.add(4,"no",Tree.contains(2));
Tree.add(23,"no",Tree.contains(4));
Tree.add(24,"yes",Tree.contains(23));
Tree.add(25,"no",Tree.contains(23));
Tree.add(26,"yes",Tree.contains(25));
Tree.add(27,"no",Tree.contains(25));
Tree.add(28,"no",Tree.contains(27));
Tree.add(29,"yes",Tree.contains(27));
Tree.add(3,"yes",Tree.contains(2));
Tree.add(5,"yes",Tree.contains(3));
Tree.add(6,"no",Tree.contains(3));
Tree.add(7,"yes",Tree.contains(6));
Tree.add(8,"no",Tree.contains(6));
Tree.add(9,"yes",Tree.contains(8));
Tree.add(17,"no",Tree.contains(9));
Tree.add(19,"yes",Tree.contains(17));
Tree.add(20,"no",Tree.contains(17));
Tree.add(21,"yes",Tree.contains(20));
Tree.add(10,"yes",Tree.contains(9));
Tree.add(11,"yes",Tree.contains(10));
Tree.add(12,"no",Tree.contains(10));
Tree.add(15,"yes",Tree.contains(12));
Tree.add(13,"no",Tree.contains(12));
Tree.add(14,"yes",Tree.contains(13));
Tree.add(16,"no",Tree.contains(13));
};
Tree = new BinarySearchTree();
ArmTheTree(Tree);
});
How can I use my implementation of the transverse function to search for a node within the binary tree? My intention is to use this tree for a true or false game.

You can use a recursive approach with an auxiliary function _contains, which looks for the value in a node in the tree and its subtrees:
...,
_contains: function(node, value) {
if (node === null) return null;
if (node.value == value) return node;
// search in left subtree
var n = this._contains(node.left, value);
if (n) return n;
// search in right subtree
return this._contains(node.right, value);
},
contains: function(value) {
return this._contains(this._root, value);
},
...
The function returns null if the node isn't found.
If you make the function add return the newly added node, you could save yourself a lot of tree-walking when building the tree and you could probably do without the contains function. Instead of:
Tree.add(1, null, null);
Tree.add(30, "no", Tree.contains(1));
Tree.add(31, "yes", Tree.contains(30));
you'd get:
var n = {};
n[1] = Tree.add(1, null, null);
n[30] = Tree.add(30, "no", n[1]);
n[31] = Tree.add(31, "yes", n[30]);