Solution Explanation

This problem involves traversing a binary tree represented in a unique way using a list of three-digit integers. Each integer encodes the depth, position, and value of a node. The goal is to calculate the sum of all root-to-leaf paths. A depth-first search (DFS) approach is the most efficient way to solve this.

Approach

1. Data Structure: We use a hash map (dictionary in Python) to store the node information. The key is the encoded node (depth * 10 + position), and the value is the node's value. This provides O(1) lookup time for node values.

2. DFS Function: A recursive dfs function explores the tree. It takes the current node's encoded representation and the current path sum as input.

3. Base Case: If the current node is not in the hash map, it means it's a leaf node or a non-existent node. In this case, the function simply returns.

4. Recursive Step:

   • The current node's value is added to the path sum.
   • The left and right children's encoded representations are calculated based on the parent's depth and position.
   • The dfs function is recursively called for the left and right children.
   • If both left and right children are not found (meaning it's a leaf node), the current path sum is added to the total sum (ans).

Time and Space Complexity

Time Complexity: O(N), where N is the number of nodes in the tree. In the worst case (a full binary tree), the DFS function visits each node once.

Space Complexity: O(N) in the worst case. This is due to the recursive call stack during the DFS traversal and the space used by the hash map to store node values. The hash map's size is at most the number of nodes.

Code Explanation (Python)

class Solution:
    def pathSum(self, nums: List[int]) -> int:
        def dfs(node, t):
            if node not in mp:
                return
            t += mp[node]
            d, p = divmod(node, 10)
            l = (d + 1) * 10 + (p * 2) - 1
            r = l + 1
            nonlocal ans
            if l not in mp and r not in mp:
                ans += t
                return
            dfs(l, t)
            dfs(r, t)
 
        ans = 0
        mp = {num // 10: num % 10 for num in nums}
        dfs(11, 0)
        return ans

• pathSum(nums): This is the main function. It initializes the ans (total sum) to 0 and creates the mp hash map. It then calls the dfs function starting from the root node (encoded as 11).

• dfs(node, t): This recursive function performs the depth-first search.

  • node: The encoded representation of the current node (e.g., 11, 21, 22, etc.).
  • t: The current path sum.
  • divmod(node, 10): This efficiently extracts the depth (d) and position (p) from the encoded node.
  • l and r: Calculate the encoded representations of the left and right children.
  • nonlocal ans: This keyword is essential because ans is modified within the nested function.
  • The base case and recursive steps are implemented as described in the approach section.

The other languages (Java, C++, Go) follow the same logic, with minor syntax variations to handle hash maps and recursive function calls. The core algorithm remains consistent across all languages.