**π‘ Problem Formulation:** When working with binary trees in Python, a common challenge is determining if the nodes at a given vertical level form a sorted sequence. This could be critical in algorithms where vertical levels reflect certain attributes that need ordering. For instance, if a binary tree represents a family hierarchy, verifying that the ages along a vertical path are increasing could be a requirement. The input would be a binary tree and a vertical level index, while the desired output is a Boolean indicating whether the vertical level is sorted.

## Method 1: Vertical Traversal and Sorting Verification

This method involves traversing the binary tree and collecting the values of each vertical level in separate lists. Once the traversal is complete, we could iterate through the list corresponding to the target vertical level and check if it is sorted. This method is straightforward and easy to implement.

Here’s an example:

def is_sorted_vertical(tree_root, vertical_level): verticals = {} def _dfs(node, x): if node: verticals.setdefault(x, []).append(node.value) _dfs(node.left, x - 1) _dfs(node.right, x + 1) _dfs(tree_root, 0) return sorted(verticals[vertical_level]) == verticals[vertical_level] # Assuming 'Node' is a class defining nodes of the binary tree # Example usage: # is_sorted_vertical(root_node, 1)

Output: `True`

if the vertical level is sorted, otherwise `False`

.

In the code snippet, the function `is_sorted_vertical()`

performs a depth-first search to populate a dictionary associating each vertical level with its values. It then simply compares the sorted list to the actual list to determine order.

## Method 2: In-Place Vertical Level Traversal

Instead of explicitly storing the entire tree representation level by level, we execute an in-place comparison while performing an in-order traversal. Only if the nodeβs horizontal distance equals the target vertical level do we keep the last seen value and compare subsequent values on-the-fly while traversing. This is memory efficient but requires careful traversal.

Here’s an example:

def is_vertical_sorted(tree_root, vertical_level): last_val = [None] def _in_order(node, level): if node is None or not last_val[0]: return True if level == vertical_level: if last_val[0] and node.value < last_val[0]: last_val[0] = False return False last_val[0] = node.value return _in_order(node.left, level - 1) and _in_order(node.right, level + 1) return _in_order(tree_root, 0) # Example usage: # is_vertical_sorted(root_node, 2)

Output: `True`

if the vertical level is sorted, `False`

otherwise.

In this code snippet, a helper function `_in_order()`

performs an in-order traversal. It effectively measures the depth and checks order as it traverses, setting a flag to False upon finding out-of-order values.

## Method 3: Iterative Tree Traversal with Level Tracking

This technique involves an iterative traversal, using a queue to keep track of nodes along with their vertical levels. As we pop nodes from the queue, we check if they belong to the target vertical level. We add a comparison check to see if the tree maintains its sorted property for that vertical level. If all checks pass, the vertical is sorted.

Here’s an example:

from collections import deque def is_sorted_vertical_iterative(tree_root, vertical_level): if not tree_root: return True queue = deque([(tree_root, 0)]) last_val = None while queue: node, level = queue.popleft() if level == vertical_level: if last_val is not None and node.value < last_val: return False last_val = node.value if node.left: queue.append((node.left, level - 1)) if node.right: queue.append((node.right, level + 1)) return True # Example usage: # is_sorted_vertical_iterative(root_node, 1)

Output: `True`

if the vertical level is sorted, `False`

otherwise.

Here the function `is_sorted_vertical_iterative()`

employs a queue to perform a level-order traversal, keeping track of vertical depths. It allows us to check for sortedness without having to store all levels.

## Method 4: Recursive Pre-order Traversal with Sortedness Verification

With recursive pre-order traversal, we traverse along the binary tree, and on each callback, we verify if the current node aligns with the sortedness condition for the vertical level we are interested in. This method is advisable if the binary tree is not balanced or not complete, as it offers a simple recursive pattern.

Here’s an example:

def check_vertical_sorted(tree_root, vertical_level): is_sorted = [True] # We are using list for mutability prev_node = [None] # Store previous node to compare with current def _check(node, level): if not node or not is_sorted[0]: return if level == vertical_level: if prev_node[0] and node.value < prev_node[0].value: is_sorted[0] = False prev_node[0] = node _check(node.left, level - 1) _check(node.right, level + 1) _check(tree_root, 0) return is_sorted[0] # Example usage: # check_vertical_sorted(root_node, 1)

Output: `True`

if the vertical level is sorted, otherwise `False`

.

Here, `check_vertical_sorted()`

uses a pre-order traversal on the binary tree. It takes a depth-first approach, comparing node values against their predecessor to ensure vertical sortedness.

## Bonus One-Liner Method 5: Using Generator Expressions

For Python enthusiasts, a one-liner is always appealing. We can leverage generator expressions and the built-in `all()`

function to perform an in-order traversal while checking the sortedness of a vertical level, all in one go.

Here’s an example:

def is_vertical_sorted_oneliner(root, vertical_level): return all(b >= a for a, b in zip( *(in_order(node, level) for level, node in enumerate_gen(root, 0) if level == vertical_level))) # Example usage and supporting functions would be required for a complete example.

Output: Expected to return `True`

if the vertical level is sorted, `False`

otherwise.

The one-liner `is_vertical_sorted_oneliner()`

compactly carries out the sortedness check. It relies heavily on Python’s generator expressions and assumes that the functions `enumerate_gen()`

and `in_order()`

are previously defined to facilitate the traversal.

## Summary/Discussion

**Method 1:**Vertical Traversal and Sorting Verification. Straightforward to implement. Requires storing all values and then checking for sortedness.**Method 2:**In-Place Vertical Level Traversal. Memory efficient. Relies on correct in-order traversal and can be tricky to implement accurately.**Method 3:**Iterative Tree Traversal with Level Tracking. Does not require recursion, which could be a benefit on highly unbalanced trees. Maintains a queue that might become large for wide trees.**Method 4:**Recursive Pre-order Traversal with Sortedness Verification. A straightforward recursive approach for trees that are not balanced or complete. Carries the overhead of the call stack.**Method 5:**Using Generator Expressions. Elegant but requires comprehensive support functions. Readability could be an issue for some.