Operations on Tree

You are given a tree with `n` nodes numbered from `0` to `n - 1` in the form of a parent array `parent` where `parent[i]` is the parent of the `ith` node. The root of the tree is node `0`, so `parent[0] = -1` since it has no parent. You want to design a data structure that allows users to lock, unlock, and upgrade nodes in the tree.

The data structure should support the following functions:

*   **Lock:** **Locks** the given node for the given user and prevents other users from locking the same node. You may only lock a node using this function if the node is unlocked.
*   **Unlock: Unlocks** the given node for the given user. You may only unlock a node using this function if it is currently locked by the same user.
*   **Upgrade****: Locks** the given node for the given user and **unlocks** all of its descendants **regardless** of who locked it. You may only upgrade a node if **all** 3 conditions are true:
    *   The node is unlocked,
    *   It has at least one locked descendant (by **any** user), and
    *   It does not have any locked ancestors.

Implement the `LockingTree` class:

*   `LockingTree(int[] parent)` initializes the data structure with the parent array.
*   `lock(int num, int user)` returns `true` if it is possible for the user with id `user` to lock the node `num`, or `false` otherwise. If it is possible, the node `num` will become **locked** by the user with id `user`.
*   `unlock(int num, int user)` returns `true` if it is possible for the user with id `user` to unlock the node `num`, or `false` otherwise. If it is possible, the node `num` will become **unlocked**.
*   `upgrade(int num, int user)` returns `true` if it is possible for the user with id `user` to upgrade the node `num`, or `false` otherwise. If it is possible, the node `num` will be **upgraded**.

**Example 1:**

**Input**
\[ "LockingTree ",  "lock ",  "unlock ",  "unlock ",  "lock ",  "upgrade ",  "lock "\]
\[\[\[-1, 0, 0, 1, 1, 2, 2\]\], \[2, 2\], \[2, 3\], \[2, 2\], \[4, 5\], \[0, 1\], \[0, 1\]\]
**Output**
\[null, true, false, true, true, true, false\]

**Explanation**
LockingTree lockingTree = new LockingTree(\[-1, 0, 0, 1, 1, 2, 2\]);
lockingTree.lock(2, 2);    // return true because node 2 is unlocked.
                           // Node 2 will now be locked by user 2.
lockingTree.unlock(2, 3);  // return false because user 3 cannot unlock a node locked by user 2.
lockingTree.unlock(2, 2);  // return true because node 2 was previously locked by user 2.
                           // Node 2 will now be unlocked.
lockingTree.lock(4, 5);    // return true because node 4 is unlocked.
                           // Node 4 will now be locked by user 5.
lockingTree.upgrade(0, 1); // return true because node 0 is unlocked and has at least one locked descendant (node 4).
                           // Node 0 will now be locked by user 1 and node 4 will now be unlocked.
lockingTree.lock(0, 1);    // return false because node 0 is already locked.

**Constraints:**

*   `n == parent.length`
*   `2 <= n <= 2000`
*   `0 <= parent[i] <= n - 1` for `i != 0`
*   `parent[0] == -1`
*   `0 <= num <= n - 1`
*   `1 <= user <= 104`
*   `parent` represents a valid tree.
*   At most `2000` calls **in total** will be made to `lock`, `unlock`, and `upgrade`.