Lessons from Path Queries

Core ideas:

$1.$ HLD works because heavy chains create contiguous segments in a linear array

$2.$ The $O(\log n)$ light-edge bound ensures we don't visit too many chains

$3.$ Any path operation (max, sum, min, gcd) works if the segment tree supports it

$4.$ Path updates (add $v$ to all nodes on path) need lazy propagation in the segment tree

HLD is the standard technique for tree path queries. Learn it well, you'll use it often.