Also here's $$$O(logN)$$$ offline but is easier and doesn't require binary lifting (though if you don't know binary lifting then you should probably learn it anyway, fairly common and appears in the recent div3).

Do a DFS traversal on the tree, maintaining an array of ancestors to the current vertex. When we enter a vertex, push its value into the array. When we exit the vertex, pop the value out of the array. The code goes something like this

dfs(u):
	append val[u] into the back of arr
	for each children v of u:
		dfs(v)
	pop the back element out of arr

Now we can process queries offline. When we reach the vertex $$$v$$$ that has queries, the array arr already contains the path from $$$1$$$ to $$$v$$$. Just binary search like online algo 1 and find the answer.