# Euler tour of Binary Tree

Given a binary tree where each node can have at most two child nodes, the task is to find the Euler tour of the binary tree. Euler tour is represented by a pointer to the topmost node in the tree. If the tree is empty, then value of root is NULL.

Examples:

Input :

Output: 1 5 4 2 4 3 4 5 1

Approach:

(2) Go to left node i.e, node 5, Euler[1]=5
(3) Go to left node i.e, node 4, Euler[2]=4
(4) Go to left node i.e, node 2, Euler[3]=2
(5) Go to left node i.e, NULL, go to parent node 4 Euler[4]=4
(6) Go to right node i.e, node 3 Euler[5]=3
(7) No child, go to parent, node 4 Euler[6]=4
(8) All child discovered, go to parent node 5 Euler[7]=5
(9) All child discovered, go to parent node 1 Euler[8]=1

Euler tour of tree has been already discussed where it can be applied to N-ary tree which is represented by adjacency list. If a Binary tree is represented by the classical structured way by links and nodes, then there need to first convert the tree into adjacency list representation and then we can find the Euler tour if we want to apply method discussed in the original post. But this increases the space complexity of the program. Here, In this post, a generalized space-optimized version is discussed which can be directly applied to binary trees represented by structure nodes.
This method :
(1) Works without the use of Visited arrays.
(2) Requires exactly 2*N-1 vertices to store Euler tour.

 // C++ program to find euler tour of binary tree #include using namespace std;    /* A tree node structure */ struct Node {     int data;     struct Node* left;     struct Node* right; };    /* Utility function to create a new Binary Tree node */ struct Node* newNode(int data) {     struct Node* temp = new struct Node;     temp->data = data;     temp->left = temp->right = NULL;     return temp; }    // Find Euler Tour void eulerTree(struct Node* root, vector &Euler) {     // store current node's data     Euler.push_back(root->data);        // If left node exists     if (root->left)      {         // traverse left subtree         eulerTree(root->left, Euler);            // store parent node's data          Euler.push_back(root->data);     }        // If right node exists     if (root->right)      {         // traverse right subtree         eulerTree(root->right, Euler);             // store parent node's data         Euler.push_back(root->data);     } }    // Function to print Euler Tour of tree void printEulerTour(Node *root) {     // Stores Euler Tour     vector Euler;         eulerTree(root, Euler);        for (int i = 0; i < Euler.size(); i++)         cout << Euler[i] << " "; }    /* Driver function to test above functions */ int main() {     // Constructing tree given in the above figure      Node* root = newNode(1);     root->left = newNode(2);     root->right = newNode(3);     root->left->left = newNode(4);     root->left->right = newNode(5);     root->right->left = newNode(6);     root->right->right = newNode(7);     root->right->left->right = newNode(8);        // print Euler Tour     printEulerTour(root);         return 0; }

Output:

1 2 4 2 5 2 1 3 6 8 6 3 7 3 1

Time Complexity: O(2*N-1) where N is number of nodes in the tree.
Auxiliary Space : O(2*N-1) where N is number of nodes in the tree.

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