Problem - Path Queries II

You are given a tree where each node has a value. Process two types of queries:

$1.$ Change the value of node s to x (update query).

$2.$ Find the maximum value on the path from node a to node b (path query).

There are up to $200$,$000$ nodes and $200$,$000$ queries mixed together. Updates and path queries can be interleaved. This requires a data structure that handles both path queries and point updates efficiently. Heavy-light decomposition with a segment tree is the standard approach.