Solution Explanation: Encoding and Decoding N-ary Tree to Binary Tree

This problem focuses on encoding an N-ary tree (a tree where each node can have more than two children) into a binary tree, and then decoding it back to the original N-ary tree. The solution utilizes a clever approach to map the N-ary tree's structure onto a binary tree's structure.

Encoding (N-ary to Binary):

The core idea is to use the left child of a binary tree node to represent the first child of the corresponding N-ary tree node and the right child of the binary tree node to represent the next sibling of the N-ary tree node. This effectively chains siblings together using the right pointers in the binary tree.

• Base Case: If the current N-ary node is null, return null (empty binary tree).
• Recursive Step: Create a binary tree node with the same value as the N-ary node.
  • If the N-ary node has children:
    • The left child of the binary tree node points to the encoded version of the first child of the N-ary node.
    • We then traverse the remaining siblings of the N-ary node, recursively encoding each sibling and linking them as right children of the previously encoded sibling in the binary tree.

Decoding (Binary to N-ary):

The decoding process mirrors the encoding. It leverages the structure created during encoding to reconstruct the N-ary tree.

• Base Case: If the current binary tree node is null, return null.
• Recursive Step: Create an N-ary tree node with the same value as the binary tree node.
  • If the binary tree node has a left child:
    • Recursively decode the left child to get the first child of the N-ary node.
    • Traverse the right children of the binary tree node, recursively decoding each to add as subsequent children of the N-ary node.

Time and Space Complexity:

Both the encoding and decoding processes visit each node in the tree exactly once. Therefore:

• Time Complexity: O(N), where N is the number of nodes in the N-ary tree.
• Space Complexity: O(N) in the worst case, due to the recursive call stack. The space used is proportional to the height of the tree. In a balanced tree, this would be O(log N), but in a skewed tree, it could be O(N). Additionally, the space used to store the N-ary tree and binary tree structures themselves is also O(N).

Code Examples (Python, Java, C++, Go, TypeScript):

The provided code snippets above demonstrate the encoding and decoding algorithms in several popular programming languages. They all follow the recursive approach described above. Note that the specific data structures (Node and TreeNode classes) might need adjustments to match the particular environment or library used. The core logic of the encoding and decoding functions remains consistent across all languages.