Challenge: Combined Techniques

Here's a challenge to test your understanding: Given a tree, support these operations:

$1.$ Add $v$ to all nodes on path from $u$ to $w$

$2.$ Query sum of values in subtree of $u$

This requires combining HLD (for path updates) with Euler tour (for subtree queries). Here's the trick: use HLD to handle path updates, but store values in Euler tour order so subtree queries are ranges. Think about how the data structures interact before looking up solutions. Build: $O(n)$. Path update: $O(\log^2 n)$. Subtree query: $O(\log n)$. Space: $O(n)$.