<p>There is an undirected connected tree with <code>n</code> nodes labeled from <code>0</code> to <code>n - 1</code> and <code>n - 1</code> edges.</p>

<p>You are given the integer <code>n</code> and the array <code>edges</code> where <code>edges[i] = [a<sub>i</sub>, b<sub>i</sub>]</code> indicates that there is an edge between nodes <code>a<sub>i</sub></code> and <code>b<sub>i</sub></code> in the tree.</p>

<p>Return an array <code>answer</code> of length <code>n</code> where <code>answer[i]</code> is the sum of the distances between the <code>i<sup>th</sup></code> node in the tree and all other nodes.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/07/23/lc-sumdist1.jpg" style="width: 304px; height: 224px;" />
<pre>
<strong>Input:</strong> n = 6, edges = [[0,1],[0,2],[2,3],[2,4],[2,5]]
<strong>Output:</strong> [8,12,6,10,10,10]
<strong>Explanation:</strong> The tree is shown above.
We can see that dist(0,1) + dist(0,2) + dist(0,3) + dist(0,4) + dist(0,5)
equals 1 + 1 + 2 + 2 + 2 = 8.
Hence, answer[0] = 8, and so on.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/07/23/lc-sumdist2.jpg" style="width: 64px; height: 65px;" />
<pre>
<strong>Input:</strong> n = 1, edges = []
<strong>Output:</strong> [0]
</pre>

<p><strong class="example">Example 3:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/07/23/lc-sumdist3.jpg" style="width: 144px; height: 145px;" />
<pre>
<strong>Input:</strong> n = 2, edges = [[1,0]]
<strong>Output:</strong> [1,1]
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= n &lt;= 3 * 10<sup>4</sup></code></li>
	<li><code>edges.length == n - 1</code></li>
	<li><code>edges[i].length == 2</code></li>
	<li><code>0 &lt;= a<sub>i</sub>, b<sub>i</sub> &lt; n</code></li>
	<li><code>a<sub>i</sub> != b<sub>i</sub></code></li>
	<li>The given input represents a valid tree.</li>
</ul>

class Solution:
    def sumOfDistancesInTree(self, n: int, edges: List[List[int]]) -> List[int]:
        

        ### let's think about this

        ## think about a subtree only edge a -> b and a < b
        ## first calculate total number of nodes at a sub tree && total sum of distance for root 
        ## run pass info to each children i.e. sdist[child] = sdist[par] - (node_cnt[child] - 1 for child) + 
        ##                                                    (total_node - node_cnt[child] - 1)

        
        adj_list = defaultdict(list)
        for node_a, node_b in edges:
            # if node_a > node_b:
            #     node_a, node_b = node_b, node_a
            adj_list[node_a].append(node_b)
            adj_list[node_b].append(node_a)

        ## root will always be 0
        def get_tree_sum(cur_node, node_cnt):
            
            ## leaf nodes
            if cur_node not in adj_list:
                return 0
            
            cur_sum = 0
            cur_tree_nodes = 0
            for next_node in adj_list[cur_node]:
                adj_list[next_node].remove(cur_node)
                if len(adj_list[next_node]) == 0:
                    adj_list.pop(next_node, None)

                if next_node in adj_list:
                    cur_sum += get_tree_sum(next_node, node_cnt)
                    cur_tree_nodes += node_cnt[next_node]
                else:
                    cur_tree_nodes += 1

            cur_sum += cur_tree_nodes
            node_cnt[cur_node] = cur_tree_nodes + 1
            return cur_sum
            
        def fill_dist_sum(dist_sum, curr_node, parent_node, node_cnt, adj_list):
            
            ## run pass info to each children i.e. 
            ## sdist[child] = sdist[par] - (node_cnt[child] - 1 for child) + 
            ##                    (total_node - node_cnt[child] - 1)

            curr_node_diff = node_cnt[0] - node_cnt[curr_node] * 2 
            dist_sum[curr_node] = dist_sum[parent_node] + curr_node_diff

            for next_node in adj_list[curr_node]:
                fill_dist_sum(dist_sum, next_node, curr_node, node_cnt, adj_list)
            return 
        
        node_cnt = [1] * n
        root_node_sum = get_tree_sum(0, node_cnt)
        # print("root node sum", root_node_sum, node_cnt, adj_list)

        dist_sum = [0] * n
        dist_sum[0] = root_node_sum
        for next_node in adj_list[0]:
            fill_dist_sum(dist_sum, next_node, 0, node_cnt, adj_list)
            # print("root next_node", next_node, dist_sum)

            

        return dist_sum

            
            





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