Has anyone solved any of these two problems using Dominator Tree?

https://cses.fi/problemset/task/1703

https://cses.fi/problemset/task/1203

Unfortunately my code is failing on a couple tests.

I'd really appreciate it if anyone provides an accepted code using Dominator Tree.

Hi, Codeforces.

Recently I have been thinking of the following task:

You are given a tree with $$$N$$$ vertices, each vertex $$$v$$$ has a value $$$a_v$$$ written on it, and you have to process following query ONLINE. given two vertices $$$u$$$ and $$$v$$$, you have to output the MEX of the values on the path from $$$u$$$ to $$$v$$$. The values are not necessarily distinct.

constrains: $$$N$$$<=1e5

If anyone can share any ideas, I will be extremely grateful.