Checking whether two paths are intersecting in a tree
This is a standard exercise.  Try drawing some pictures with the different cases, and you'll see how to formulate a criterion for when they overlap, in terms of the lowest common ancestor.

Just computing $LCA(u_1,v_1)$ isn't gonna be enough, because that only gives you information about the first path, and doesn't depend at all on the second path.  So try computing the LCA of some other pair of nodes.

Since this looks like an exercise, I'm not going to solve your problem for you, but I'll describe for you how you can approach this problem so you can solve it yourself.


Every path falls into one of two categories: either it goes down (i.e., $v_1$ is a descendant of $u_1$); or it goes up and then down (i.e., $u_1$ and $v_1$ are descendants of some common ancestor, $w_1 = LCA(u_1,v_1)$, where $w_1 \ne u_1$ and $w_1 \ne v_1$).  So, there are two possibilities for the first path, and two possibilities for the second path: four cases in all.
You can quickly classify each path, put it into these categories, and find the endpoints and the common ancestor.  (Exercise for you: How?)


Now, you have four cases.  For each case, draw some pictures and work out the criteria under which the two paths intersect.  It's straightforward.

For instance, let's take the first of the four cases.  Suppose both paths go only down: $u_1$ is an ancestor of $v_1$, and $u_2$ is an ancestor of $v_2$.  Can you figure out how to tell whether they overlap?  Some tips: If $LCA(u_1,u_2)=u_1$, what can you conclude?  If $LCA(u_1,u_2)=u_2$, what can you conclude?  If $LCA(u_1,u_2)$ is something else, then what can you conclude?

Now do similar reasoning for each of the four cases.

Note: I am also assuming that you have a rooted tree; otherwise, the problem is not well-defined (as the LCA is not defined in non-rooted trees).

My thanks to @babou for his helpful comments and feedback on an earlier version of this answer.