Pick The Dynamic Tree

A forest changes over time: link, cut, and after each change a query for the sum of values on the path from $u$ to $v$ . Queries are online. What do you build?

A rollback DSU over time. Paths are exactly what its union structure aggregates, and link and cut are $O(\log n)$ amortized.

A link/cut tree. Paths are exactly what preferred paths aggregate, and link and cut are $O(\log n)$ amortized.

Heavy light decomposition. Paths are exactly what its chains aggregate, and link and cut cost $O(\log n)$ per rebuild.

An Euler tour tree. Paths are exactly what a difference of two tour prefixes gives, and link and cut are $O(\log n)$ .