Encode N-ary Tree to Binary Tree

Problem Statement

The challenge is to design a unique algorithm capable of encoding an N-ary tree into a binary tree and subsequently decoding that binary tree back into the original N-ary tree structure. An N-ary tree is characterized by each node having up to N children, while a binary tree limits each node to a maximum of two children. The provided algorithm should neither impose constraints on the method of encoding/decoding nor rely on any particular serialized format, though examples are provided for illustrative purposes where the N-ary tree's input serialization is exhibited in their level order traversal, delineating each group of children with a null value. The primary objective here is to maintain the structural integrity of the N-ary tree throughout this conversion process.

Examples

Example 1

Input:

root = [1,null,3,2,4,null,5,6]

Output:

[1,null,3,2,4,null,5,6]

Example 2

Input:

root = [1,null,2,3,4,5,null,null,6,7,null,8,null,9,10,null,null,11,null,12,null,13,null,null,14]

Output:

[1,null,2,3,4,5,null,null,6,7,null,8,null,9,10,null,null,11,null,12,null,13,null,null,14]

Example 3

Input:

root = []

Output:

[]

Constraints

• The number of nodes in the tree is in the range [0, 104].
• 0 <= Node.val <= 104
• The height of the n-ary tree is less than or equal to 1000
• Do not use class member/global/static variables to store states. Your encode and decode algorithms should be stateless.

Approach and Intuition

The complex task of encoding an N-ary tree to a binary tree and decoding it back hinges on mapping the multiple children of N-ary nodes into the binary format while ensuring all original tree data and structure are preserved and can be accurately reconstructed.

1. Encoding Technique:

   • You may choose to utilize the left child of the binary tree to represent the first child of the N-ary node, and the right child of this binary node to point to the next sibling in the N-ary node. This technique effectively turns the array representation with scattered null values as separators into a zigzag pattern across the binary tree.
   • Another anticipated approach could involve additional marking or use of metadata within nodes to maintain order and relationship of children when transferred over to a binary format.

2. Decoding Technique:

   • The decoding process must mirror the encoding logic carefully, unraveling the two-child constraint of the binary format to revive the original multi-children format of the N-ary tree. This should be efficiently handled to revert any marks or metadata used during encoding.
   • Decode operations need to manage and maintain correct child-to-parent relations and ensure all N children nodes are recovered precisely to their original form.

3. Consideration for Constraints:

   • The encoding and decoding functions should operate without storing state between them—both should be stateless, relying only on input data and not capturing any data outside the input scope (like global/static variables).
   • The maximum allowed tree size and node values must be handled gracefully, with the algorithm providing consistent performance irrespective of tree height, adhering to the mentioned constraints of node count and node value.

By handling these steps judiciously, the encoded binary tree would retain the essential structure and data of the original N-ary tree, making an accurate reconstruction possible via decoding. This method fundamentally pivots on creative data management and structural mappings between the two tree types.

Solutions

class Codec {
    
    // Function to convert an n-ary tree into a binary tree
    public TreeNode encode(Node root) {
        if (root == null) return null;
    
        TreeNode binaryRoot = new TreeNode(root.val);
        if (!root.children.isEmpty()) {
            binaryRoot.left = this.encode(root.children.get(0));
        }
    
        TreeNode current = binaryRoot.left;
        for (int i = 1; i < root.children.size(); i++) {
            current.right = this.encode(root.children.get(i));
            current = current.right;
        }
    
        return binaryRoot;
    }
    
    // Function to convert a binary tree back into an n-ary tree
    public Node decode(TreeNode root) {
        if (root == null) return null;
    
        Node nAryRoot = new Node(root.val, new ArrayList<>());
        TreeNode current = root.left;
    
        while (current != null) {
            nAryRoot.children.add(this.decode(current));
            current = current.right;
        }
    
        return nAryRoot;
    }
}