EDIT For followup Question:

Thanks for the clarifications :)

But, One follow-up question is that, In this problem https://oj.leetcode.com/problems/triangle/, you can reach a node in different ways, but it can be solvable with DFS.

public class Solution {
    int[][] memo;
    public int minimumTotal(List<List<Integer>> triangle) {
        memo = new int[triangle.size()][triangle.size()];
        for(int i = 0; i < triangle.size(); i++) {
            Arrays.fill(memo[i], Integer.MIN_VALUE);
        }
        return minimumTotal(triangle,0,0);
    }
    public int minimumTotal(List<List<Integer>> triangle, int i, int j) {
        if (i == triangle.size() -1)
           return triangle.get(i).get(j); // base case
        if(memo[i][j] != INTEGER.MIN_VALUE) {
            return memo[i][j];
        }
        int sum0 = minimumTotal(triangle, i+1, j);
        int sum1 = minimumTotal(triangle, i+1, j+1);
        int res = Math.min(sum0, sum1) + triangle.get(i).get(j);
        memo[i][j] = res;
        return res;
    }   
}

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Original post:

Why we cannot use DFS to find the shortest distance on the graph/mazes, while it can be used on the tree.

How can I formally prove that even a modified DFS doesn't work.

For ex: we can solve this one ( https://oj.leetcode.com/problems/triangle/) easily with DFS.

Thanks for your help !!!