Isomorphism in N-ary Trees
Given two N-ary trees having M nodes each. Also, given their edges and their roots respectively. The task is to check if they are isomorphic trees or not. If both the trees are isomorphic then print “Yes” else print “No”.
Examples:
Input: M = 9, Root Node of tree-1: 1, Root Node of tree-2: 3
Edges of Tree-1: { (1, 3), (3, 4), (3, 5), (1, 8), (8, 9), (1, 2), (2, 6), (2, 7) }
Edges of Tree-2: { (3, 1), (1, 2), (1, 5), (3, 6), (6, 7), (3, 4), (4, 8), (4, 9) }
Output: YES
Input: M = 9, Root Node of tree-1: 6, Root Node of tree-2: 7
Edges of Tree-1: {(1, 3),(1, 2), (1, 8), (3, 4), (3, 5), (8, 9), (2, 6), (2, 7)}
Edges of Tree-2: {(1, 2), (1, 5), (3, 1), (3, 4), (4, 8), (4, 9), (6, 3), (7, 6)}
Output: NO
Approach:
The idea is to find out the canonical form of both the trees and comparing them. The leaf node will return “()” to its subsequent upper levels.
Below is the example showing the process of finding the canonical form.
Below is the implementation of the above approach:
CPP
// C++ program for the above approach#include <bits/stdc++.h>using namespace std;// To create N-ary treemap<int, vector<int> > tree;// Function which will accept the root// of the tree and its parent (which// is initially "-1") and will return// the canonical form of the treestring ConvertCanonical(int vertex, int parent){ // In this string vector we will // store canonical form of out // current node and its subtree vector<string> child; // Loop to the neighbours of // current vertex for (auto neighbour : tree[vertex]) { // This is to prevent us from // visiting the parent again if (neighbour == parent) continue; // DFS call neighbour of our // current vertex & it will // pushback the subtree-structure // of this neighbour into our // child vector. child.push_back(ConvertCanonical( neighbour, vertex)); } // These opening and closing // brackets are for the // current node string str = "("; // Sorting function will re-order // the structure of subtree of // the current vertex in a // shortest-subtree-first manner. // Hence we can // now compare the two tree // structures effectively sort(child.begin(), child.end()); for (auto j : child) str += j; // Append the subtree structure // and enclose it with "(" <subtree> // ")" the opening and closing // brackets of current vertex str += ")"; // return the subtree-structure // of our Current vertex return str;}// Function to add edgesvoid addedge(int a, int b){ tree[a].push_back(b); tree[b].push_back(a);}// Driver codeint main(){ // Given N-ary Tree 1 addedge(1, 3); addedge(1, 2); addedge(1, 5); addedge(3, 4); addedge(4, 8); addedge(4, 9); addedge(3, 6); addedge(6, 7); // Function Call to convert Tree 1 // into canonical with 3 is the root // and the parent of root be "-1" string tree1 = ConvertCanonical(3, -1); // Clearing our current tree // before taking input of // next tree tree.clear(); // Given N-ary Tree 2 addedge(1, 3); addedge(3, 4); addedge(3, 5); addedge(1, 8); addedge(8, 9); addedge(1, 2); addedge(2, 6); addedge(2, 7); // Function Call to convert Tree 2 // into canonical string tree2 = ConvertCanonical(1, -1); // Check if canonical form of both // tree are equal or not if (tree1 == tree2) cout << "YES" << endl; else cout << "NO" << endl; return 0;} |
Python3
# Python3 program for the above approach# To create N-ary treetree=dict()# Function which will accept the root# of the tree and its parent (which# is initially "-1") and will return# the canonical form of the treedef ConvertCanonical(vertex, parent): # In this string vector we will # store canonical form of out # current node and its subtree child=[] # Loop to the neighbours of # current vertex for neighbour in tree[vertex] : # This is to prevent us from # visiting the parent again if (neighbour == parent): continue # DFS call neighbour of our # current vertex & it will # pushback the subtree-structure # of this neighbour into our # child vector. child.append(ConvertCanonical( neighbour, vertex)) # These opening and closing # brackets are for the # current node s = "(" # Sorting function will re-order # the structure of subtree of # the current vertex in a # shortest-subtree-first manner. # Hence we can # now compare the two tree # structures effectively child.sort() for j in child: s += j # Append the subtree structure # and enclose it with "(" <subtree> # ")" the opening and closing # brackets of current vertex s += ")" # return the subtree-structure # of our Current vertex return s# Function to add edgesdef addedge(a, b): if a in tree: tree[a].append(b) else: tree[a]=[b,] if b in tree: tree[b].append(a) else: tree[b]=[a,]# Driver codeif __name__=='__main__': # Given N-ary Tree 1 addedge(1, 3) addedge(1, 2) addedge(1, 5) addedge(3, 4) addedge(4, 8) addedge(4, 9) addedge(3, 6) addedge(6, 7) # Function Call to convert Tree 1 # into canonical with 3 is the root # and the parent of root be "-1" tree1 = ConvertCanonical(3, -1) # Clearing our current tree # before taking input of # next tree tree.clear() # Given N-ary Tree 2 addedge(1, 3) addedge(3, 4) addedge(3, 5) addedge(1, 8) addedge(8, 9) addedge(1, 2) addedge(2, 6) addedge(2, 7) # Function Call to convert Tree 2 # into canonical tree2 = ConvertCanonical(1, -1) # Check if canonical form of both # tree are equal or not if (tree1 == tree2): print("YES") else: print("NO") |
Output:
YES
Time Complexity: O(E * log(E)), where E is the number of edges.
Auxiliary Space: O(E)
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