#### Approach #1: Subtree Sum and Count [Accepted]

Intuition

Let ans be the returned answer, so that in particular ans[x] be the answer for node x.

Naively, finding each ans[x] would take time (where is the number of nodes in the graph), which is too slow. This is the motivation to find out how ans[x] and ans[y] are related, so that we cut down on repeated work.

Let's investigate the answers of neighboring nodes and . In particular, say is an edge of the graph, that if cut would form two trees (containing ) and (containing ). Then, as illustrated in the diagram, the answer for in the entire tree, is the answer of on "x@X", plus the answer of on "y@Y", plus the number of nodes in "#(Y)". The last part "#(Y)" is specifically because for any node z in Y, dist(x, z) = dist(y, z) + 1.

By similar reasoning, the answer for in the entire tree is ans[y] = x@X + y@Y + #(X). Hence, for neighboring nodes and , ans[x] - ans[y] = #(Y) - #(X).

Algorithm

Root the tree. For each node, consider the subtree of that node plus all descendants. Let count[node] be the number of nodes in , and stsum[node] ("subtree sum") be the sum of the distances from node to the nodes in .

We can calculate count and stsum using a post-order traversal, where on exiting some node, the count and stsum of all descendants of this node is correct, and we now calculate count[node] += count[child] and stsum[node] += stsum[child] + count[child].

This will give us the right answer for the root: ans[root] = stsum[root].

Now, to use the insight explained previously: if we have a node parent and it's child child, then these are neighboring nodes, and so ans[child] = ans[parent] - count[child] + (N - count[child]). This is because there are count[child] nodes that are 1 easier to get to from child than parent, and N-count[child] nodes that are 1 harder to get to from child than parent. Using a second, pre-order traversal, we can update our answer in linear time for all of our nodes.

Complexity Analysis

• Time Complexity: , where is the number of nodes in the graph.

• Space Complexity: .

Analysis written by: @awice.