Need to work on recursive solution to solve minimum path sum - depth first search, work on base case, and then recurrence formula using Math.Min for various children.

MinimumPathSum_Practice.cs

using System;

class Solution
{
  public class Node
  {
    public int Cost {get; set; }
    public Node[] Children {get; set;}
   // public Node parent;     
    
    public Node(int number)
    {
      Cost = number; 
    }
  }
    
  public static int getCheapestCost(Node rootNode) // node with 0
  {
    if(rootNode == null) // false 
    {
      return 0; 
    }
    
    var rootValue = rootNode.Cost; 
    
    // base case 
    if(rootNode.Children == null) // on the leaves of tree, true 
    {
      return rootValue; 
    }    
   
    var cheapestValue = Int32.MaxValue; 
    
    foreach(var item in rootNode.Children) // node with 5, with 3, with 6
    {
      var childPath = getCheapestCost(item); // 9, 13, 7
      cheapestValue = childPath < cheapestValue? childPath : cheapestValue; // 9, 7
    }
    
    return rootValue + cheapestValue; // 0 + 7 = 7
  }

  static void Main(string[] args)
  {
    var node0 = new Node(0); 
    
    // from left to right 
    var node5 = new Node(5); 
    var node3 = new Node(3); 
    var node6 = new Node(6); 
    
    var node4 = new Node(4); 
    var node02 = new Node(0);
    var node10 = new Node(10); 
    
    var nodeSecond1 = new Node(1); 
    var nodeSecond5 = new Node(5); 
    
    // second level 
    node0.Children = new Node[]{node5, node3, node6}; 
    
    // third level 
    node5.Children = new Node[]{node4}; 
    node3.Children = new Node[]{node02}; 
    
    node6.Children = new Node[]{nodeSecond1, nodeSecond5}; 
    
    // fourth level 
    node02.Children = new Node[]{node10}; 
    
    Console.WriteLine(getCheapestCost(node0));  //0+5+4, 0+3+0+10, 0+6+1, 0+6+5   
  }
}
/*
time complexity O(n) - n is total nodes in the tree
space complexity is O(n)-O(1)

*/