i am hitting TLE error and please check if the given solution is the solution for the porblem like look for all edge cases explain me throughly and rectlify the code or better it 
```
# Question 1: Locking the Tree of Space
# You have a world map represented as an M-Ary tree (sample tree below):

# You need to define three operations on it:
# lock(X, uid)
# unlock(X, uid)
# upgradeLock(X, uid)
# Where X is the name of a node in the tree (which is unique), and uid is the user performing the operation.
# Definitions for the Operations:
# Lock(X, uid):
# Lock takes exclusive access on the subtree rooted at X. Once lock(X, uid) succeeds:
# lock(A, anyUserId) should fail (returns false), where A is a descendant of X.
# lock(B, anyUserId) should fail (returns false), where X is a descendant of B.
# Lock operation cannot be performed on a node that is already locked, i.e., lock(X, anyUserId) should fail (returns false).
# Unlock(X, uid):
# Unlock reverts what was done by the lock operation. It can only be called on the same node on which user uid had called a lock before. Returns true if successful.
# UpgradeLock(X, uid):
# This helps the user uid upgrade their lock to an ancestor node. It is only possible if the node X already has locked descendants and all of them are locked by the same user uid. Upgrade should fail if there is any node which is a descendant of X that is locked by a different user. A successful upgrade will "lock" the node. UpgradeLock must not violate the consistency model required by Lock/Unlock.
# Notes:
# The number of nodes in the tree, N, is very large, so optimize the time complexity for the above algorithms.
# Input Format:
# The first line contains the number of nodes in the tree (N).
# The second line contains the number of children per node (m in an m-ary tree).
# The third line contains the number of queries (Q).
# Next N lines contain the NodeName (string) in the m-Ary Tree.
# The next Q lines contain queries in the format: OperationType NodeName UserId.
# Operation Type:
# 1 for Lock
# 2 for Unlock
# 3 for UpgradeLock
# NodeName: The name of any node (unique) in the m-Ary Tree.
# UserId: An integer representing a unique user.
# Example Input:
# 7
# 2
# 3
# World
# Asia
# Africa
# China
# India
# South Africa
# Egypt
# 1 China 9
# 2 India 9
# 3 Asia 9


# This input represents a 2-ary tree with 7 nodes as follows:
#            World
#            /     \
#       Asia        Africa
#      /    \       /     \
#  China  India  South Africa  Egypt


# Additional Notes:
# The tree is always fully balanced.
# Constraints:
# (1 < N < 5 \times 10^5)
# (1 < m < 30)
# (1 < Q < 5 \times 10^5)
# (1 < \text{length of NodeName} < 20)
# Optimize the time complexity:
# Lock: (O(\log_m N))
# Unlock: (O(\log_m N))
# UpgradeLock: (O(\text{numberOfLockedNodes} \times \log_m N))
# The lock operation on an already locked node should fail.
# Once UpgradeLock(X, uid) succeeds on X, it is equivalent to X being locked by uid. As per the definition, Lock(A/B, anyUser) should fail, and Unlock(X, uid) should also work.
# Upgrade lock operation on a node having no locked descendants should fail, and upgrade lock on an already locked node should also fail.


import math

class Node:
    def init(self, val):
        self.val = val          # Node value (name)
        self.uid = None         # User ID who locked the node
        self.parent = None      # Reference to parent node
        self.children = []      # List of child nodes
        self.isLocked = False   # Lock status
        self.lockedDescendantCount = 0  # Number of locked descendants

class Tree:
    def init(self, n, m, node_list, queries):
        self.node_list = []
        self.node_map = {}
        
        # Create node instances and map node names to instances
        for name in node_list:
            node = Node(name)
            self.node_list.append(node)
            self.node_map[name] = node
        
        # Assign children to each node based on m-ary tree properties
        for i in range(n):
            for j in range(m):
                child_index = m * i + j + 1
                if child_index < n:
                    self.node_list[i].children.append(self.node_list[child_index])
                    self.node_list[child_index].parent = self.node_list[i]
        
        self.root = self.node_list[0]
        
        # Parse queries
        self.processed_queries = [query.strip().split() for query in queries]
        
    def lock(self, node, uid):
        # Check if node is already locked
        if node.isLocked:
            return False
        
        # Check if any ancestors are locked
        if self.isAncestorLocked(node):
            return False
        
        # Check if any descendants are locked
        if node.lockedDescendantCount > 0:
            return False
        
        # Lock the node
        node.isLocked = True
        node.uid = uid
        
        # Update ancestors' lockedDescendantCount
        parent = node.parent
        while parent:
            parent.lockedDescendantCount += 1
            parent = parent.parent
        
        return True

    def unlock(self, node, uid):
        # Check if node is locked by uid
        if not node.isLocked or node.uid != uid:
            return False
        
        # Unlock the node
        node.isLocked = False
        node.uid = None
        
        # Update ancestors' lockedDescendantCount
        parent = node.parent
        while parent:
            parent.lockedDescendantCount -= 1
            parent = parent.parent
        
        return True

    def upgrade(self, node, uid):
        # Check if node is already locked
        if node.isLocked:
            return False
        
        # Check if any ancestors are locked
        if self.isAncestorLocked(node):
            return False
        
        # Collect all locked descendants
        locked_descendants = []
        if not self.collectLockedDescendants(node, uid, locked_descendants):
            return False
        
        if not locked_descendants:
            # No locked descendants to upgrade
            return False
        
        # Unlock all locked descendants
        for descendant in locked_descendants:
            self.unlock(descendant, uid)
        
        # Lock the node
        return self.lock(node, uid)
    
    def isAncestorLocked(self, node):
        # Traverse up to check if any ancestor is locked
        current = node.parent
        while current:
            if current.isLocked:
                return True
            current = current.parent
        return False
    
    def collectLockedDescendants(self, node, uid, locked_nodes):
        # Collect locked descendants locked by uid
        result = True
        if node.isLocked:
            if node.uid == uid:
                locked_nodes.append(node)
            else:
                # Locked by a different user
                return False
        for child in node.children:
            if not self.collectLockedDescendants(child, uid, locked_nodes):
                result = False
        return result

    def process_queries(self):
        results = []
        for query in self.processed_queries:
            operation, node_name, uid = query
            node = self.node_map[node_name]
            if operation == "1":
                result = self.lock(node, uid)
            elif operation == "2":
                result = self.unlock(node, uid)
            elif operation == "3":
                result = self.upgrade(node, uid)
            else:
                result = False  # Invalid operation
            results.append(result)
        return results
```
in poirper sstructure