Construct a Complete N-ary Tree from given Postorder Traversal
Last Updated :
23 Jul, 2025
Given an array arr[] of size M that contains the post-order traversal of a complete N-ary tree, the task is to generate the N-ary tree and print its preorder traversal.
A complete tree is a tree where all the levels of the tree are completely filled, except may be for the last level but the nodes in the last level are as left as possible.
Examples:
Input: arr[] = {5, 6, 7, 2, 8, 9, 3, 4, 1}, N = 3
Output:
The tree structure for above example
Input: arr[] = {7, 8, 9, 2, 3, 4, 5, 6, 1}, N = 5
Output:
Complete structure for the 2nd example
Approach: The approach to the problem is based on the following facts about the complete N-ary tree:
Say any subtree of the complete N-ary tree has a height H. So there are total H+1 levels numbered from 0 to H.
Total number of nodes in first H levels are N0 + N1 + N2 + . . . + NH-1 = (NH - 1)/(N - 1).
The maximum possible nodes in the last level is NH.
So if the last level of the subtree has at least NH nodes, then total nodes in the last level of the subtree is (NH - 1)/(N - 1) + NH
The height H can be calculated as ceil[logN( M*(N-1) +1)] - 1 because
N-ary complete binary tree there can have at max (NH+1 - 1)/(N - 1)] nodes
Since the given array contains the post-order traversal, the last element of the array will always be the root node. Now based on the above observation the remaining array can be partitioned into a number of subtrees of that node.
Follow the steps mentioned below to solve the problem:
- The last element of the array is the root of the tree.
- Now break the remaining array into subarrays which represent the total number of nodes in each subtree.
- Each of these subtrees surely has a height of H-2 and based on the above observation if any subtree has more than (NH-1 - 1)/(N - 1) nodes then it has a height of H-1.
- To calculate the number of nodes in subtrees follow the below cases:
- If the last level has more than NH-1 nodes, then all levels in this subtree are full and the subtree has (NH-1-1)/(N-1) + NH-1 nodes.
- Otherwise, the subtree will have (NH-1-1)/(N-1) + L nodes where L is the number of nodes in the last level.
- L can be calculated as L = M - (NH - 1)/(N - 1).
- To generate each subarray repeatedly apply the 2nd to 4th steps adjusting the size (M) and height (H) accordingly for each subtree.
- Return the preorder traversal of the tree formed in this way
Follow the illustration below for a better understanding
Illustration:
Consider the example: arr[] = {5, 6, 7, 2, 8, 9, 3, 4, 1}, N = 3.
So height (H) = ceil[ log3(9*2 + 1) ] - 1 = ceil[log319] - 1 = 3 - 1 = 2 and L = 9 - (32 - 1)/2 = 5.
1st step: So the root = 1
1st step of forming the tree
2nd step: The remaining array will be broken into subtrees. For the first subtree H = 2.
5 > 32-1 i.e., L > 3. So the last level is completely filled and the number of nodes in the first subtree = (3-1)/2 + 3 = 4 [calculated using the formulae shown above].
The first subtree contains element = {5, 6, 7, 2}. The remaining part is {8, 9, 3, 4}
The root of the subtree = 2.
Now when the subtree for 5, 6, and 7 are calculated using the same method they don't have any children and are the leaf nodes themselves.
2nd step of generating the tree
3rd step: Now L is updated to 2.
2 < 3. So using the above formula, number of nodes in the second subtree is 1 + 2 = 3.
So the second subtree have elements {8, 9, 3} and the remaining part is {4} and 3 is the root of the second subtree.
3rd step of generating the tree
4th step: L = 0 now and the only element of the subtree is {4} itself.
Final structure of tree
After this step the tree is completely built.
Below is the implementation of the above approach:
C++
// C++ code to implement the approach
#include <bits/stdc++.h>
using namespace std;
// Node Class
template <typename T> class Node {
public:
Node(T data);
// Get the first child of the Node
Node* get_first_child() const;
// Get the Next Sibling of The node
Node* get_next_sibling() const;
// Sets the next sibling of the node
void set_next_sibling(Node* node);
// Sets the next child of the node
void set_next_child(Node* node);
// Returns the data the node contains
T get_data();
private:
T data;
// We use the first child/next sibling representation
// to represent the Tree
Node* first_child;
Node* next_sibling;
};
// Using template for generic usage
template <typename T> Node<T>::Node(T data)
{
first_child = NULL;
next_sibling = NULL;
this->data = data;
}
// Function to get the first child
template <typename T>
Node<T>* Node<T>::get_first_child() const
{
return first_child;
}
// Function to get the siblings
template <typename T>
Node<T>* Node<T>::get_next_sibling() const
{
return next_sibling;
}
// Function to set next sibling
template <typename T>
void Node<T>::set_next_sibling(Node<T>* node)
{
if (next_sibling == NULL) {
next_sibling = node;
}
else {
next_sibling->set_next_sibling(node);
}
}
// Function to get the data
template <typename T> T Node<T>::get_data() { return data; }
// Function to set the child node value
template <typename T>
void Node<T>::set_next_child(Node<T>* node)
{
if (first_child == NULL) {
first_child = node;
}
else {
first_child->set_next_sibling(node);
}
}
// Function to construct the tree
template <typename T>
Node<T>* Construct(T* post_order_arr, int size, int k)
{
Node<T>* Root = new Node<T>(post_order_arr[size - 1]);
if (size == 1) {
return Root;
}
int height_of_tree
= ceil(log2(size * (k - 1) + 1) / log2(k)) - 1;
int nodes_in_last_level
= size - ((pow(k, height_of_tree) - 1) / (k - 1));
int tracker = 0;
while (tracker != (size - 1)) {
int last_level_nodes
= (pow(k, height_of_tree - 1)
> nodes_in_last_level)
? nodes_in_last_level
: (pow(k, height_of_tree - 1));
int nodes_in_next_subtree
= ((pow(k, height_of_tree - 1) - 1) / (k - 1))
+ last_level_nodes;
Root->set_next_child(
Construct(post_order_arr + tracker,
nodes_in_next_subtree, k));
tracker = tracker + nodes_in_next_subtree;
nodes_in_last_level
= nodes_in_last_level - last_level_nodes;
}
return Root;
}
// Function to print the preorder traversal
template <typename T> void preorder(Node<T>* Root)
{
if (Root == NULL) {
return;
}
cout << Root->get_data() << " ";
preorder(Root->get_first_child());
preorder(Root->get_next_sibling());
}
// Driver code
int main()
{
int M = 9, N = 3;
int arr[] = { 5, 6, 7, 2, 8, 9, 3, 4, 1 };
// Function call
preorder(Construct(arr, M, N));
return 0;
}
// Java code to implement the approach
import java.io.*;
import java.util.*;
class GFG
{
// Function to calculate the
// log base 2 of an integer
public static int log2(int N)
{
// calculate log2 N indirectly
// using log() method
int result = (int)(Math.log(N) / Math.log(2));
return result;
}
// Node Class
public static class Node {
int data;
// We use the first child/next sibling
// representation to represent the Tree
Node first_child;
Node next_sibling;
// Initializing Node using Constructor
public Node(int data)
{
this.first_child = null;
this.next_sibling = null;
this.data = data;
}
// Function to get the first child
public Node get_first_child()
{
return this.first_child;
}
// Function to get the siblings
public Node get_next_sibling()
{
return this.next_sibling;
}
// Function to set the next sibling
public void set_next_sibling(Node node)
{
if (this.next_sibling == null) {
this.next_sibling = node;
}
else {
this.next_sibling.set_next_sibling(node);
}
}
// Function to get the data
public int get_data() { return this.data; }
// Function to set the child node values
public void set_next_child(Node node)
{
if (this.first_child == null) {
this.first_child = node;
}
else {
this.first_child.set_next_sibling(node);
}
}
}
public static void main(String[] args)
{
int M = 9, N = 3;
int[] arr = { 5, 6, 7, 2, 8, 9, 3, 4, 1 };
// Function call
preorder(Construct(arr, 0, M, N));
}
// Function to print the preorder traversal
public static void preorder(Node Root)
{
if (Root == null) {
return;
}
System.out.println(Root.get_data() + " ");
preorder(Root.get_first_child());
preorder(Root.get_next_sibling());
}
// Function to construct the tree
public static Node Construct(int[] post_order_arr,
int tracker, int size,
int k)
{
Node Root = new Node(post_order_arr[tracker+size - 1]);
if (size == 1) {
return Root;
}
int height_of_tree
= (int)Math.ceil(log2(size * (k - 1) + 1)
/ log2(k))
- 1;
int nodes_in_last_level
= size
- (((int)Math.pow(k, height_of_tree) - 1)
/ (k - 1));
int x=tracker;
while (tracker != (size - 1)) {
int last_level_nodes
= ((int)Math.pow(k, height_of_tree - 1)
> nodes_in_last_level)
? nodes_in_last_level
: ((int)Math.pow(k,
height_of_tree - 1));
int nodes_in_next_subtree
= (((int)Math.pow(k, height_of_tree - 1)
- 1)
/ (k - 1))
+ last_level_nodes;
Root.set_next_child(Construct(
post_order_arr, tracker, nodes_in_next_subtree, k));
tracker = tracker + nodes_in_next_subtree;
nodes_in_last_level
= nodes_in_last_level - last_level_nodes;
}
return Root;
}
}
# Python code to implement the approach
import math
# Node Class
class Node:
# We use the first child/next sibling representation to represent the Tree
def __init__(self, data):
self.data = data
self.first_child = None
self.next_sibling = None
# Function to get the first child
def get_first_child(self):
return self.first_child
# Function to get the siblings
def get_next_sibling(self):
return self.next_sibling
# Function to set next sibling
def set_next_sibling(self, node):
if self.next_sibling is None:
self.next_sibling = node
else:
self.next_sibling.set_next_sibling(node)
# Function to set the child node value
def set_next_child(self, node):
if self.first_child is None:
self.first_child = node
else:
self.first_child.set_next_sibling(node)
# Function to get the data
def get_data(self):
return self.data
# Function to construct the tree
def Construct(post_order_arr, size, k):
Root = Node(post_order_arr[size - 1])
if size == 1:
return Root
height_of_tree = math.ceil(
math.log(size * (k - 1) + 1, 2) / math.log(k, 2)) - 1
nodes_in_last_level = size - ((pow(k, height_of_tree) - 1) / (k - 1))
tracker = 0
while tracker != (size - 1):
if pow(k, height_of_tree-1) > nodes_in_last_level:
last_level_nodes = int(nodes_in_last_level)
else:
last_level_nodes = int((pow(k, height_of_tree - 1)))
nodes_in_next_subtree = int(
((pow(k, height_of_tree - 1) - 1) / (k - 1)) + last_level_nodes)
Root.set_next_child(
Construct(post_order_arr[tracker:], nodes_in_next_subtree, k))
tracker = tracker + nodes_in_next_subtree
nodes_in_last_level = nodes_in_last_level - last_level_nodes
return Root
# Function to print the preorder traversal
def preorder(root):
if root == None:
return
print(root.get_data(), end=" ")
preorder(root.get_first_child())
preorder(root.get_next_sibling())
# Driver code
if __name__ == '__main__':
M = 9
N = 3
arr = [5, 6, 7, 2, 8, 9, 3, 4, 1]
# Function call
preorder(Construct(arr, M, N))
# This code is contributed by Tapesh(tapeshdua420)
// C# code to implement the approach
using System;
using System.Collections.Generic;
using System.Linq;
// Node Class
class Program {
public class Node {
int data;
// We use the first child/next sibling
// representation to represent the Tree
Node first_child;
Node next_sibling;
public Node(int data)
{
this.first_child = null;
this.next_sibling = null;
this.data = data;
}
// Function to get the first child
public Node get_first_child()
{
return this.first_child;
}
// Function to get the siblings
public Node get_next_sibling()
{
return this.next_sibling;
}
// Function to set next sibling
public void set_next_sibling(Node node)
{
if (this.next_sibling == null) {
this.next_sibling = node;
}
else {
this.next_sibling.set_next_sibling(node);
}
}
// Function to get the data
public int get_data() { return this.data; }
// Function to set the child node value
public void set_next_child(Node node)
{
if (this.first_child == null) {
this.first_child = node;
}
else {
this.first_child.set_next_sibling(node);
}
}
}
// Driver code
public static void Main(string[] args)
{
int M = 9, N = 3;
int[] arr = { 5, 6, 7, 2, 8, 9, 3, 4, 1 };
preorder(Construct(arr, M, N));
}
// Function to print the preorder traversal
public static void preorder(Node Root)
{
if (Root == null) {
return;
}
Console.Write(Root.get_data() + " ");
preorder(Root.get_first_child());
preorder(Root.get_next_sibling());
}
// Function to construct the tree
public static Node Construct(int[] post_order_arr,
int size, int k)
{
Node Root = new Node(post_order_arr[size - 1]);
if (size == 1) {
return Root;
}
int height_of_tree = (int)Math.Ceiling(
(double)(Math.Log(size * (k - 1) + 1, 2)
/ Math.Log(k, 2))
- 1);
// Console.WriteLine(height_of_tree);
int nodes_in_last_level
= size
- (((int)Math.Pow(k, height_of_tree) - 1)
/ (k - 1));
int tracker = 0;
while (tracker != (size - 1)) {
int last_level_nodes
= ((int)Math.Pow(k, height_of_tree - 1)
> nodes_in_last_level)
? nodes_in_last_level
: ((int)Math.Pow(k,
height_of_tree - 1));
int nodes_in_next_subtree
= (int)((Math.Pow(k, height_of_tree - 1)
- 1)
/ (k - 1))
+ last_level_nodes;
Root.set_next_child(
Construct((post_order_arr.Skip(tracker))
.Cast<int>()
.ToArray(),
nodes_in_next_subtree, k));
tracker = tracker + nodes_in_next_subtree;
nodes_in_last_level
= nodes_in_last_level - last_level_nodes;
}
return Root;
}
}
// This Code is contributed by Tapesh(tapeshdua420)
// JavaScript code for the above approach
// Node Class
class Node {
// We use the first child/next sibling representation to represent the Tree
constructor(data) {
this.data = data;
this.firstChild = null;
this.nextSibling = null;
}
// Function to get the first child
getFirstChild() {
return this.firstChild;
}
// Function to get the siblings
getNextSibling() {
return this.nextSibling;
}
// Function to set next sibling
setNextSibling(node) {
if (this.nextSibling === null) {
this.nextSibling = node;
} else {
this.nextSibling.setNextSibling(node);
}
}
// Function to set the child node value
setNextChild(node) {
if (this.firstChild === null) {
this.firstChild = node;
} else {
this.firstChild.setNextSibling(node);
}
}
// Function to get the data
getData() {
return this.data;
}
}
// Function to construct the tree
function construct(postOrderArr, size, k) {
const root = new Node(postOrderArr[size - 1]);
if (size === 1) {
return root;
}
const heightOfTree = Math.ceil((Math.log((size * (k - 1)) + 1, 2) / Math.log(k, 2))) - 1;
let nodesInLastLevel = size - ((Math.pow(k, heightOfTree) - 1) / (k - 1));
let tracker = 0;
while (tracker !== size - 1) {
let lastLevelNodes;
if (Math.pow(k, heightOfTree - 1) > nodesInLastLevel) {
lastLevelNodes = Math.floor(nodesInLastLevel);
} else {
lastLevelNodes = Math.floor(Math.pow(k, heightOfTree - 1));
}
const nodesInNextSubtree = Math.floor(((Math.pow(k, heightOfTree - 1) - 1) / (k - 1)) + lastLevelNodes);
root.setNextChild(construct(postOrderArr.slice(tracker, tracker + nodesInNextSubtree), nodesInNextSubtree, k));
tracker += nodesInNextSubtree;
nodesInLastLevel -= lastLevelNodes;
}
return root;
}
// Function to print the preorder traversal
function preorder(root) {
if (root === null) {
return;
}
console.log(root.getData() + " ");
preorder(root.getFirstChild());
preorder(root.getNextSibling());
}
// Driver code
const M = 9;
const N = 3;
const arr = [5, 6, 7, 2, 8, 9, 3, 4, 1];
// Function call
preorder(construct(arr, M, N));
// This code is contributed by Potta Lokesh
Time Complexity: O(M)
Auxiliary Space: O(M) for building the tree
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