4718：The LCIS on the Tree

For a sequence S₁, S₂, ... , S_N, and a pair of integers (i, j), if 1 <= i <= j <= N and S_i < S_i+1 < S_i+2 < ... < S_j-1 < S_j , then the sequence S_i, S_i+1, ... , S_j is a CIS (Continuous Increasing Subsequence). The longest CIS of a sequence is called the LCIS (Longest Continuous Increasing Subsequence).
Now we consider a tree rooted at node 1. Nodes have values. We have Q queries, each with two nodes u and v. You have to find the shortest path from u to v. And write down each nodes' value on the path, from u to v, inclusive. Then you will get a sequence, and please show us the length of its LCIS.

The first line has a number T (T <= 10) , indicating the number of test cases.
For each test case, the first line is a number N (N <= 10⁵), the number of nodes in the tree.
The second line comes with N numbers v1, v2, v3 ... , v_N, describing the value of node 1 to node N. (1 <= v_i <= 10⁹)
The third line comes with N - 1 numbers p₂, p₃, p₄ ... , p_N, describing the father nodes of node 2 to node N. Node 1 is the root and will have no father.
Then comes a number Q, it is the number of queries. (Q <= 10⁵)
For next Q lines, each with two numbers u and v. As described above.

For test case X, output "Case #X:" at the first line.
Then output Q lines, each with an answer to the query.
There should be a blank line *BETWEEN* each test case.