July 2024, Week 3: July 15th - July 21st

The central concept of this solution is to leverage a hash map (dictionary) to efficiently construct the binary tree in a single pass through the descriptions. This approach exploits the property that each node in a binary tree has a unique value, allowing us to use these values as keys in our hash map.

• Node Storage: Each unique node is stored in a hash map, allowing access when connecting parent-child relationships.

• Root Identification: The root is identified as the only parent node that doesn't appear as a child in any description.

• Tree Construction: The tree is built incrementally by processing each description and connecting nodes based on the given parent-child relationships.

1. Initialization:

   # Create a map of child values to their corresponding nodes node_map = {child: BinaryTreeNode(child) for _, child, _ in descriptions}

   This line creates a dictionary node_map where each key is a child value from the descriptions, and the corresponding value is a new BinaryTreeNode with that value. This initializes all nodes that appear as children in the descriptions.

2. Root Initialization and Tree Construction:

   root = None for parent_value, child_value, is_left_child in descriptions: if parent_value not in node_map: # Create root node (the only parent value not present as a child) root = node_map[parent_value] = BinaryTreeNode(parent_value)

   This loop iterates through each description. If a parent value is encountered that's not in the node_map, it means this node hasn't been seen as a child before, so it must be the root. We create this node and assign it to both root and node_map[parent_value].

3. Child Assignment:

   # Assign child node to appropriate side of parent if is_left_child: node_map[parent_value].left = node_map[child_value] else: node_map[parent_value].right = node_map[child_value]

   For each description, we assign the child node to either the left or right of the parent node based on the is_left_child value. Both parent and child nodes are accessed from the node_map in time.

4. Result Return:

   return root

   After processing all descriptions, we return the root node of the constructed binary tree.

Time Complexity:

• , where is the number of descriptions (which is equal to the number of edges in the tree). We iterate through the descriptions once to create the node_map, and then again to construct the tree. Both operations are , resulting in a total time complexity of .

Space Complexity:

• , where is the number of descriptions. The node_map stores one entry for each unique node in the tree. In the worst case, this could be up to nodes ( child nodes plus one root node that only appears as a parent). Therefore, the space complexity is .

Input:

Step-by-Step Walkthrough:

1. Initialization:

   • The function starts by creating a node_map dictionary. This map uses child values as keys and BinaryTreeNode objects as values.

   • Initial node_map:

     node_map = { 58: BinaryTreeNode(58), 39: BinaryTreeNode(39), 45: BinaryTreeNode(45), 43: BinaryTreeNode(43), 41: BinaryTreeNode(41), 52: BinaryTreeNode(52) }

   • root is initialized as None.

2. Main Algorithm Process:

   • Iteration 1:

     • Processing [52, 58, 0]

     • 52 is in node_map, so no new node is created.

     • 58 is assigned as the right child of 52 because is_left_child is 0.

     • Current tree: 52 -> 58 (right)

   • Iteration 2:

     • Processing [41, 39, 1]

     • 41 is in node_map, so no new node is created.

     • 39 is assigned as the left child of 41 because is_left_child is 1.

     • Current tree: 52 -> 58 (right), 41 -> 39 (left)

   • Iteration 3:

     • Processing [52, 45, 1]

     • 52 already exists in node_map.

     • 45 is assigned as the left child of 52 (is_left_child is 1).

     • Current tree: 52 -> 58 (right), 45 (left), 41 -> 39 (left)

   • Iteration 4:

     • Processing [41, 43, 0]

     • 41 already exists in node_map.

     • 43 is assigned as the right child of 41 (is_left_child is 0).

     • Current tree: 52 -> 58 (right), 45 (left), 41 -> 39 (left), 43 (right)

   • Iteration 5:

     • Processing [45, 41, 1]

     • 45 already exists in node_map.

     • 41 is assigned as the left child of 45 (is_left_child is 1).

     • Current tree: 52 -> 58 (right), 45 (left), 45 -> 41 (left), 41 -> 39 (left), 43 (right)

   • Iteration 6:

     • Processing [60, 52, 1]

     • 60 is not in node_map, so a new BinaryTreeNode(60) is created and set as root because it is the only parent node not present as a child.

     • 52 is assigned as the left child of 60.

     • Final tree: 60 -> 52 (left), 52 -> 58 (right), 45 (left), 45 -> 41 (left), 41 -> 39 (left), 43 (right)

3. Loop Termination:

   • The loop terminates after processing all descriptions in the input list.

   • The final state of the tree is complete, with 60 as the root node.

4. Visual Aid: Here's a summary table of the iterations:

   Iteration	Parent	Child	Child Position	Parent Node Status
   1	52	58	Right	Existed
   2	41	39	Left	Existed
   3	52	45	Left	Existed
   4	41	43	Right	Existed
   5	45	41	Left	Existed
   6	60	52	Left	Created

5. Result Calculation/Final Steps:

   • After processing all descriptions, the function has constructed the entire binary tree.

   • The root variable holds the reference to the root node of the tree, which is 60.

   • The function returns this root node, representing the entire constructed binary tree.

The final tree structure looks like this:

The central concept of this solution is to leverage the Lowest Common Ancestor (LCA) algorithm to find the meeting point of the paths from the start node to the destination node. This approach effectively breaks down the problem into three main steps: finding the LCA, constructing paths from the LCA to both the start and destination nodes, and then combining these paths to form the final result.

• Lowest Common Ancestor (LCA): The solution first identifies the LCA of the start and destination nodes. This is the deepest node in the tree that is an ancestor of both nodes.

• Path Construction: Once the LCA is found, the algorithm constructs two separate paths: one from the LCA to the start node, and another from the LCA to the destination node.

• Path Combination: The final step involves combining these paths. The path to the start node is converted to a series of 'U' moves (to go up to the LCA), while the path to the destination is kept as is (to go down from the LCA).

1. Defining Helper Functions:

   def findCommonAncestor(node: BinaryTreeNode) -> Optional[BinaryTreeNode]: # ... (function body) def findPath(current_node: BinaryTreeNode, target_value: int, path: List[str]) -> bool: # ... (function body)

   These helper functions are defined within the main function to encapsulate specific tasks. findCommonAncestor is responsible for locating the Lowest Common Ancestor (LCA) of the start and destination nodes, while findPath constructs the path from a given node to a target node.

2. Finding the Lowest Common Ancestor:

   def findCommonAncestor(node: BinaryTreeNode) -> Optional[BinaryTreeNode]: # Base case: if node is None or is one of the target nodes if node is None or node.val == start_value or node.val == dest_value: return node # Recurse on left and right subtrees left_result = findCommonAncestor(node.left) right_result = findCommonAncestor(node.right) # If both subtrees return a node, this is the LCA if left_result and right_result: return node # Otherwise, return the non-None result (if any) return left_result or right_result

   This function uses a recursive approach to find the LCA. It traverses the tree, returning the node itself if it's None or matches either the start or destination value. If both left and right subtrees return a non-None value, the current node is the LCA. Otherwise, it propagates the non-None result upwards.

3. Constructing Paths:

   def findPath(current_node: BinaryTreeNode, target_value: int, path: List[str]) -> bool: # Recursively find the path to target node, # appending directions to 'path' if current_node.val == target_value: return True if current_node.left: path.append('L') if findPath(current_node.left, target_value, path): return True path.pop() if current_node.right: path.append('R') if findPath(current_node.right, target_value, path): return True path.pop() return False

   This function uses depth-first search to find the path from a given node to the target node. It appends 'L' or 'R' to the path list as it traverses left or right. If the target is found, it returns True, propagating the success upwards. If a path doesn't lead to the target, it backtracks by removing the last direction (path.pop()).

4. Identifying the LCA and Finding Paths:

   lowest_common_ancestor = findCommonAncestor(root) path_to_start = [] path_to_dest = [] findPath(lowest_common_ancestor, start_value, path_to_start) findPath(lowest_common_ancestor, dest_value, path_to_dest)

   Here, we first find the LCA of the start and destination nodes. Then, we construct two paths: one from the LCA to the start node and another from the LCA to the destination node. This approach minimizes unnecessary traversals by starting from the LCA rather than the root for both paths.

5. Constructing the Final Path:

   # Construct the final path: 'U's to go up to LCA, initial_path = 'U' * len(path_to_start) # then path down to destination final_path = ''.join(path_to_dest) return initial_path + final_path

   The final path is constructed in two parts:

   1. initial_path: A string of 'U's with length equal to the path from LCA to start node. This represents the upward movement from the start node to the LCA.

   2. final_path: The path from LCA to the destination node, converted from a list of directions to a string.

   These two parts are concatenated to form the complete path from start to destination via the LCA.

Time Complexity:

• , where is the number of nodes in the binary tree. The time complexity is linear because in the worst case, we may need to visit each node once during the LCA search. The later path constructions from the LCA to the start and destination nodes are also linear in the height of the tree, which is at most in a skewed tree.

Space Complexity:

• , where is the height of the binary tree. The space complexity is determined by the maximum depth of the recursion stack during the DFS traversals. In the worst case of a skewed tree, this could be , but for a balanced tree, it would be . Additionally, we store the paths to the start and destination nodes, but these are also bounded by the height of the tree.

Input:

A visual representation of the binary tree with the green node representing the start node (3) and the red node representing the destination node (6) is shown below:

Step-by-Step Walkthrough:

1. Initialization:

   • The function is called with root (a binary tree with the structure shown in the input), start_value = 3, and dest_value = 6.

   • Two helper functions are defined within getDirections1: findCommonAncestor and findPath.

2. Finding Lowest Common Ancestor (LCA):

   • The findCommonAncestor function is called with the root node (value 5).

   • Iteration 1 (node 5):

     • Recursively calls findCommonAncestor on left child (1) and right child (2).

   • Iteration 2 (node 1):

     • Recursively calls findCommonAncestor on left child (3).

   • Iteration 3 (node 3):

     • Matches start_value, returns node 3.

   • Iteration 4 (node 2):

     • Recursively calls findCommonAncestor on left child (6).

   • Iteration 5 (node 6):

     • Matches dest_value, returns node 6.

   • The LCA is determined to be node 5, as both left and right subtrees return non-None values.

   • An image of the tree with the LCA (node 5) highlighted is shown below:

     

3. Finding Path to Start:

   • findPath is called with lowest_common_ancestor (node 5) and start_value (3).

   • Iteration 1 (node 5):

     • Moves left to node 1.

   • Iteration 2 (node 1):

     • Moves left to node 3.

   • Iteration 3 (node 3):

     • Matches start_value, path is complete.

   • Final path_to_start: ['L', 'L']

4. Finding Path to Destination:

   • findPath is called with lowest_common_ancestor (node 5) and dest_value (6).

   • Iteration 1 (node 5):

     • Tries left subtree (node 1), backtracks.

     • Moves right to node 2.

   • Iteration 2 (node 2):

     • Moves left to node 6.

   • Iteration 3 (node 6):

     • Matches dest_value, path is complete.

   • Final path_to_dest: ['R', 'L']

5. Constructing Final Path:

   • initial_path: 'U' * len(path_to_start) = 'UU'

   • final_path: ''.join(path_to_dest) = 'RL'

   • result: initial_path + final_path = 'UURL'

6. Visual Aid - Path Construction Summary:

   Component	Value	Length
   Initial Path (U's)	UU	2
   Final Path	RL	2
   Result	UURL	4

7. Result Calculation:

   • The final result 'UURL' represents:

     • 'UU': Move up twice from the start node (3) to reach the LCA (5).

     • 'R': Move right from the LCA to node 2.

     • 'L': Move left from node 2 to reach the destination node (6).

The function returns 'UURL' as the shortest path from node 3 to node 6 in the given binary tree.

The central concept of this solution is to leverage full path construction from the root to both the start and destination nodes, followed by a comparison to identify the point of divergence. This approach effectively breaks down the problem into three main steps: finding complete paths, identifying the common prefix, and constructing the final path.

• Full Path Construction: The solution begins by finding the complete paths from the root to both the start node and the destination node. This is done using a depth-first search (DFS) approach.

• Common Prefix Identification: Once both paths are constructed, the algorithm identifies the longest common prefix between these paths. This common prefix represents the shared path from the root to the lowest common ancestor (LCA) of the start and destination nodes.

• Path Combination: The final step involves constructing the result by combining:

  1. A series of 'U' moves to go up from the start node to the LCA.

  2. The remaining path from the LCA to the destination node.

1. Defining the Helper Function:

   def findPath(current_node: BinaryTreeNode, target_value: int, path: List[str]) -> bool: # ... (function body)

   This helper function is defined within the main function to encapsulate the task of finding a path from a given node to a target node. It's a crucial part of the solution as it will be used to find paths from the root to both the start and destination nodes.

2. Path Finding Implementation:

   def findPath(current_node: BinaryTreeNode, target_value: int, path: List[str]) -> bool: # Recursively find the path to target node, # appending directions to 'path' if current_node.val == target_value: return True if current_node.left: path.append('L') if findPath(current_node.left, target_value, path): return True path.pop() if current_node.right: path.append('R') if findPath(current_node.right, target_value, path): return True path.pop() return False

   This function uses a depth-first search approach to find the path from the current node to the target node. It appends 'L' or 'R' to the path list as it traverses left or right. If the target is found, it returns True, propagating the success upwards. If a path doesn't lead to the target, it backtracks by removing the last direction (path.pop()). This method ensures that only the correct path remains in the path list when the function completes successfully.

3. Finding Paths from Root:

   # Find paths from root to start and destination nodes path_to_start = [] path_to_dest = [] findPath(root, start_value, path_to_start) findPath(root, dest_value, path_to_dest)

   Here, we use the findPath function to construct two complete paths: one from the root to the start node and another from the root to the destination node. This approach differs from the previous solution as it finds full paths from the root instead of paths from the Lowest Common Ancestor (LCA).

4. Identifying the Common Prefix:

   # Find the length of the common prefix in both paths common_prefix_length = 0 while (common_prefix_length < len(path_to_start) and common_prefix_length < len(path_to_dest) and path_to_start[common_prefix_length] == path_to_dest[common_prefix_length]): common_prefix_length += 1

   This step identifies the length of the common prefix between the two paths. The common prefix represents the shared path from the root to the LCA of the start and destination nodes. By comparing the paths element by element, we can determine where they diverge, which implicitly identifies the LCA without a separate LCA finding algorithm.

5. Constructing the Final Path:

   # Construct the final path: 'U's to go up to LCA, initial_path = 'U' * (len(path_to_start) - common_prefix_length) # then path down to destination final_path = ''.join(path_to_dest[common_prefix_length:]) return initial_path + final_path

   The final path is constructed in two parts:

   1. initial_path: A string of 'U's with length equal to the number of steps from the start node to the LCA. This is calculated by subtracting the length of the common prefix from the length of the path to the start node.

   2. final_path: The path from LCA to the destination node, obtained by slicing the path_to_dest list from the index where the common prefix ends.

   These two parts are concatenated to form the complete path from start to destination via the LCA. This method efficiently constructs the required path without explicitly finding the LCA node.

Time Complexity:

• , where is the number of nodes in the binary tree. The time complexity is linear because in the worst case, we need to traverse the entire tree twice - once to find the path to the start node and once for the destination node. The later comparison of paths and construction of the final result are linear in the height of the tree, which is at most in a skewed tree.

Space Complexity:

• , where is the height of the binary tree. The space complexity is determined by two factors:

  1. The maximum depth of the recursion stack during the DFS traversals, which is at most the height of the tree.

  2. The storage of the two paths (path_to_start and path_to_dest), which are also bounded by the height of the tree. In the worst case of a skewed tree, this could be , but for a balanced tree, it would be .

Input:

A visual representation of the binary tree with the green node representing the start node (3) and the red node representing the destination node (6) is shown below:

Step-by-Step Walkthrough:

1. Initialization:

   • The function is called with root (a binary tree with the structure shown in the input), start_value = 3, and dest_value = 6.

   • Two empty lists are initialized: path_to_start = [] and path_to_dest = [].

2. Finding Path to Start:

   • The findPath function is called with the root node (value 5) and start_value (3).

   • Iteration 1 (node 5):

     • Moves left to node 1.

   • Iteration 2 (node 1):

     • Moves left to node 3.

   • Iteration 3 (node 3):

     • Matches start_value, path is complete.

   • Final path_to_start: ['L', 'L']

   • An image of the path from the root to the start node (3) is shown below:

     

3. Finding Path to Destination:

   • The findPath function is called with the root node (value 5) and dest_value (6).

   • Iteration 1 (node 5):

     • Tries left subtree (node 1), backtracks.

     • Moves right to node 2.

   • Iteration 2 (node 2):

     • Moves left to node 6.

   • Iteration 3 (node 6):

     • Matches dest_value, path is complete.

   • Final path_to_dest: ['R', 'L']

   • An image of the path from the root to the destination node (6) is shown below:

     

4. Finding Common Prefix:

   • The algorithm compares path_to_start and path_to_dest to find the common prefix.

   • In this case, there is no common prefix as the paths diverge from the root.

   • Common Prefix Length: 0

5. Constructing Final Path:

   • initial_path: 'U' * (len(path_to_start) - common_prefix_length) = 'UU'

   • final_path: ''.join(path_to_dest[common_prefix_length:]) = 'RL'

   • result: initial_path + final_path = 'UURL'

6. Visual Aid - Path Construction Summary:

   Component	Value	Length
   Path to Start	LL	2
   Path to Destination	RL	2
   Common Prefix		0
   Initial Path (U's)	UU	2
   Final Path	RL	2
   Result	UURL	4

7. Result Calculation:

   • The final result 'UURL' represents:

     • 'UU': Move up twice from the start node (3) to reach the root (5).

     • 'R': Move right from the root to node 2.

     • 'L': Move left from node 2 to reach the destination node (6).

The function returns 'UURL' as the shortest path from node 3 to node 6 in the given binary tree.

The central concept of this solution is to leverage a depth-first traversal (DFS) combined with a set data structure to efficiently delete nodes and construct a forest from a binary tree. This approach allows for a single pass through the tree while simultaneously building the resulting forest.

• Depth-First Traversal: The solution uses a post-order traversal to process the tree from bottom to top, ensuring that child nodes are handled before their parents.

• Set-based Deletion: By converting the to_delete list into a set, the solution achieves lookup time for checking if a node should be deleted.

• Forest Construction: As nodes are processed, the algorithm builds the forest by adding roots of subtrees that become disconnected due to node deletions.

• In-place Modification: The original tree structure is modified in-place, with deleted nodes being replaced by None of their parent's references.

1. Initialization:

   if not root: return [] forest_roots = [] nodes_to_delete = set(to_delete)

   The function starts by handling the edge case of an empty tree, returning an empty list. It then initializes an empty list forest_roots to store the roots of the resulting forest, and converts the to_delete list into a set for lookup time.

2. Helper Function Definition:

   def processNodesAndBuildForest(current_node: BinaryTreeNode) -> ( Optional)[BinaryTreeNode]:

   A helper function is defined to encapsulate the main logic of processing nodes, deleting specified ones, and building the forest. This function will be called recursively to traverse the tree.

3. Null Node Handling:

   if not current_node: return None

   Within the helper function, null nodes are immediately returned as None. This serves as the base case for the recursion and handles leaf nodes.

4. Recursive Processing of Child Nodes:

   current_node.left = processNodesAndBuildForest(current_node.left) current_node.right = processNodesAndBuildForest(current_node.right)

   The function recursively processes the left and right children of the current node. This is a post-order traversal, ensuring that child nodes are processed before their parent. The results of these recursive calls are assigned back to the current node's left and right pointers, potentially modifying the tree structure.

5. Node Deletion and Forest Building:

   if current_node.val in nodes_to_delete: # Add children of deleted node as new roots in the forest if current_node.left: forest_roots.append(current_node.left) if current_node.right: forest_roots.append(current_node.right) return None # Node is deleted by returning None return current_node

   If the current node's value is in the set of nodes to delete, its children (if they exist) are added to the forest_roots list as new roots. The node itself is "deleted" by returning None, which will be assigned to its parent's corresponding child pointer in step 4. If the node is not to be deleted, it's returned as is.

6. Main Tree Processing:

   # Process the entire tree root = processNodesAndBuildForest(root)

   The helper function is called on the root of the tree, starting the traversal and deletion process. The returned value is assigned back to root, potentially modifying the original tree if the root was deleted.

7. Result Calculation/Return:

   # Add the root to `forest_roots` if it wasn't deleted if root: forest_roots.append(root) return forest_roots

   After processing, if the root still exists (i.e., it wasn't deleted), it's added to the forest_roots list. Finally, the forest_roots list is returned, containing all the roots of the resulting forest.

Time Complexity:

• , where is the number of nodes in the binary tree. The algorithm performs a single depth-first traversal of the tree, visiting each node exactly once. The use of a set for to_delete ensures that checking if a node should be deleted is an O(1) operation. Therefore, the overall time complexity is linear with respect to the number of nodes.

Space Complexity:

• , where is the height of the tree and is the number of nodes to delete.

  • The recursion stack in the depth-first traversal can go as deep as the height of the tree, contributing to the space complexity.

  • The nodes_to_delete set contains elements, contributing to the space complexity.

  • The forest_roots list, in the worst case where all nodes are disconnected, could contain up to elements. However, this is bounded by (deleted nodes' children plus potentially the original root), so it doesn't add to the overall complexity.

  • Since in the worst case, can be equal to (for a skewed tree), and , we can simplify the space complexity to .

Input:

Here's a visual representation of the binary tree with the nodes to delete highlighted in red:

Step-by-Step Walkthrough:

1. Initialization:

   • Create an empty list forest_roots = [] to store the roots of the resulting forest.

   • Convert to_delete into a set nodes_to_delete = {1, 2, 5} for efficient lookup.

2. Main Algorithm Process:

   • Root Level (Node 1):

     • Start processing the root node with value 1.

     • Recursively process the left child (Node 2).

   • Left Subtree (Node 2):

     • Process Node 2.

     • Left and right children are None, so return None for both.

     • Node 2 is in nodes_to_delete, so return None.

     • Back to Node 1, its left child is now None.

     

   • Back to Root Level (Node 1):

     • Process the right child (Node 3).

   • Right Subtree (Node 3):

     • Process Node 3.

     • Recursively process the left child (Node 4).

   • Left Subtree of Node 3 (Node 4):

     • Process Node 4.

     • Left and right children are None, so return None for both.

     • Node 4 is not in nodes_to_delete, so return Node 4.

     • Back to Node 3, its left child remains Node 4.

   • Back to Node 3:

     • Process the right child (Node 5).

   • Right Subtree of Node 3 (Node 5):

     • Process Node 5.

     • Recursively process the left child (Node 6).

   • Left Subtree of Node 5 (Node 6):

     • Process Node 6.

     • Left and right children are None, so return None for both.

     • Node 6 is not in nodes_to_delete, so return Node 6.

     • Back to Node 5, its left child remains Node 6.

   • Back to Node 5:

     • Process the right child (Node 7).

   • Right Subtree of Node 5 (Node 7):

     • Process Node 7.

     • Left and right children are None, so return None for both.

     • Node 7 is not in nodes_to_delete, so return Node 7.

     • Back to Node 5, its right child remains Node 7.

   • Back to Node 5:

     • Node 5 is in nodes_to_delete.

     • Add Node 6 and Node 7 to forest_roots.

     • Return None to Node 3.

   • Back to Node 3:

     • Right child is now None.

     • Node 3 is not in nodes_to_delete, so return Node 3.

   • Back to Root Level (Node 1):

     • Left child is None, right child is Node 3.

     • Node 1 is in nodes_to_delete.

     • Add Node 3 to forest_roots.

     • Return None.

3. Final Processing:

   • The root (Node 1) was deleted, so it's not added to forest_roots.

4. Visual Aid: Tree Structure and Deletion Process Here's a summary of the node processing:

   Node Value	Action	Children Added to Forest
   2	Deleted	None
   4	Kept	N/A
   6	Kept	N/A
   7	Kept	N/A
   5	Deleted	6, 7
   3	Kept	N/A
   1	Deleted	3

5. Result Calculation:

   • The forest_roots list is built during the traversal.

   • The final forest_roots contains [6, 7, 3], representing the roots of the resulting forest.

The algorithm effectively deletes nodes 1, 2, and 5, creating a forest of three trees rooted at nodes 3, 6, and 7. Node 4 remains a child of Node 3 in the resulting structure.

The central concept of this solution is to leverage a single-pass depth-first search (DFS) with explicit root tracking to efficiently delete nodes and construct a forest from a binary tree. This approach allows for simultaneous node deletion and forest construction in a single traversal.

• Single-Pass DFS: The solution uses a pre-order traversal to process the tree from top to bottom, making decisions about node deletion and forest construction as it goes.

• Root Tracking: By passing an is_root boolean parameter in the recursive calls, the algorithm can determine whether a node should be considered as a potential new root in the forest.

• Set-based Deletion: Similar to the previous approach, a set is used for lookup time when checking if a node should be deleted.

• In-place Modification: The original tree structure is modified in-place, with deleted nodes being replaced by None of their parent's references.

1. Initialization:

   if not root: return [] nodes_to_delete = set(to_delete) forest_roots = []

   The function begins by handling the edge case of an empty tree, returning an empty list. It then converts the to_delete list into a set for lookup time and initializes an empty list forest_roots to store the roots of the resulting forest.

2. Helper Function Definition:

   def processNodesAndBuildForest(current_node: Optional[BinaryTreeNode], is_root: bool) -> Optional[BinaryTreeNode]:

   A helper function is defined to encapsulate the main logic of processing nodes, handling deletions, and building the forest. This function takes two parameters: the current node and a boolean is_root to track potential new roots.

3. Null Node Handling:

   if not current_node: return None

   Within the helper function, null nodes are immediately returned as None. This serves as the base case for the recursion and handles leaf nodes or already deleted subtrees.

4. Deletion Check:

   is_deleted = current_node.val in nodes_to_delete

   The function checks if the current node's value is in the set of nodes to delete, storing the result in is_deleted for later use.

5. Root Addition to Forest:

   if is_root and not is_deleted: forest_roots.append(current_node)

   If the current node is a root (either the original root or a child of a deleted node) and it's not meant to be deleted, it's added to the forest_roots list. This step is crucial for building the resulting forest.

6. Recursive Processing of Child Nodes:

   current_node.left = processNodesAndBuildForest(current_node.left, is_deleted) current_node.right = processNodesAndBuildForest(current_node.right, is_deleted)

   The function recursively processes the left and right children of the current node. Note that is_deleted is passed as the is_root parameter for the child calls. This is because if the current node is deleted, its children become new roots in the forest. The results of these recursive calls are assigned back to the current node's left and right pointers, potentially modifying the tree structure.

7. Node Deletion:

   return None if is_deleted else current_node

   If the current node is to be deleted (is_deleted is True), the function returns None, effectively removing the node from the tree. Otherwise, it returns the current node unchanged.

8. Main Tree Processing:

   processNodesAndBuildForest(root, True)

   The helper function is called on the root of the tree, with is_root set to True, starting the traversal and deletion process.

9. Result Return:

   return forest_roots

   Finally, the forest_roots list is returned, containing all the roots of the resulting forest.

Time Complexity:

• , where is the number of nodes in the binary tree. The algorithm performs a single depth-first traversal of the tree, visiting each node exactly once. The use of a set for nodes_to_delete ensures that checking if a node should be deleted is an O(1) operation. Therefore, the overall time complexity is linear with respect to the number of nodes.

Space Complexity:

• , where is the height of the tree and is the number of nodes to delete.

  • The recursion stack in the depth-first traversal can go as deep as the height of the tree, contributing to the space complexity.

  • The nodes_to_delete set contains elements, contributing to the space complexity.

  • The forest_roots list, in the worst case where all nodes are disconnected, could contain up to elements. However, this is bounded by (deleted nodes' children plus potentially the original root), so it doesn't add to the overall complexity.

  • Since in the worst case, can be equal to (for a skewed tree), and , we can simplify the space complexity to .

Input:

Here's a visual representation of the binary tree with the nodes to delete highlighted in red:

Step-by-Step Walkthrough:

1. Initialization:

   • Create a set nodes_to_delete = {1, 2, 5} for efficient lookup.

   • Initialize an empty list forest_roots = [] to store the roots of the resulting forest.

2. Main Algorithm Process:

   • Root Level (Node 1):

     • Current node: 1, is_root: True

     • 1 is in nodes_to_delete, so it will be deleted

     • Process left child (Node 2) with is_root = True

   • Left Subtree (Node 2):

     • Current node: 2, is_root: True

     • 2 is in nodes_to_delete, so it will be deleted

     • Process left child (None) with is_root = True

     • Process right child (None) with is_root = True

     • Return None (Node 2 is deleted)

   • Back to Root Level (Node 1):

     • Left child is now None

     • Process right child (Node 3) with is_root = True

   • Right Subtree (Node 3):

     • Current node: 3, is_root: True

     • 3 is not in nodes_to_delete, so add it to forest_roots

     • Process left child (Node 4) with is_root = False

   • Left Subtree of Node 3 (Node 4):

     • Current node: 4, is_root: False

     • 4 is not in nodes_to_delete

     • Process left child (None) with is_root = False

     • Process right child (None) with is_root = False

     • Return Node 4 (not deleted)

   • Back to Node 3:

     • Left child remains Node 4

     • Process right child (Node 5) with is_root = False

   • Right Subtree of Node 3 (Node 5):

     • Current node: 5, is_root: False

     • 5 is in nodes_to_delete, so it will be deleted

     • Process left child (Node 6) with is_root = True

   • Left Subtree of Node 5 (Node 6):

     • Current node: 6, is_root: True

     • 6 is not in nodes_to_delete, so add it to forest_roots

     • Process left child (None) with is_root = False

     • Process right child (None) with is_root = False

     • Return Node 6 (not deleted)

   • Back to Node 5:

     • Left child remains Node 6

     • Process right child (Node 7) with is_root = True

   • Right Subtree of Node 5 (Node 7):

     • Current node: 7, is_root: True

     • 7 is not in nodes_to_delete, so add it to forest_roots

     • Process left child (None) with is_root = False

     • Process right child (None) with is_root = False

     • Return Node 7 (not deleted)

   • Back to Node 5:

     • Right child remains Node 7

     • Return None (Node 5 is deleted)

   • Back to Node 3:

     • Right child is now None

     • Return Node 3 (not deleted)

   • Back to Root Level (Node 1):

     • Right child remains Node 3

     • Return None (Node 1 is deleted)

3. Visual Aid: Tree Structure and Deletion Process Here's a summary of the node processing:

   Node Value	Action	Was Root	Children
   2	Deleted	True	Left: None, Right: None
   4	Kept	False	Left: None, Right: None
   6	Kept	True	Left: None, Right: None
   7	Kept	True	Left: None, Right: None
   5	Deleted	False	Left: 6, Right: 7
   3	Kept	True	Left: 4, Right: None
   1	Deleted	True	Left: None, Right: 3

4. Result Calculation:

   • The forest_roots list contains [3, 6, 7], representing the roots of the three trees in the resulting forest.

Final Output:

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