Is there a simple solution to the below problem in $$$O(1)$$$ per query with reasonable, such as $$$O(NlogN)$$$, preprocessing?

**Problem:** Given two nodes $$$u$$$ and $$$v$$$ in a tree, find the second node on the path from $$$u$$$ to $$$v$$$.

If $$$u$$$ is not an ancestor of $$$v$$$ this is easy. If $$$u$$$ is an ancestor of $$$v$$$ though then the answer is one of the children of $$$u$$$. You could compute the answer with jump-pointers in $$$O(log N)$$$ per query, but is there any easy way to find it in $$$O(1)$$$ per query?