Post-order traversal always visits the left subtree before the right subtree because this order mirrors mathematical convention and standardised tree traversal logic. Visiting the left subtree first aligns with how binary expressions are typically written and read in infix form, where the left operand comes before the right operand. This order becomes particularly important in expression trees, where operator precedence and operand order must be preserved. However, in some non-expression-based applications, such as tree deletion or structural analysis, the exact order of left versus right subtree might not affect the final result, provided both subtrees are processed before the root. That said, adhering to the standard left-right-root approach ensures consistency and compatibility with existing algorithms and tools. In academic and professional contexts, this conventional order is assumed unless explicitly specified otherwise, and deviating from it without justification can lead to incorrect outputs or inconsistencies when integrating with other software components or evaluation mechanisms.

In general trees, which can have more than two children, post-order traversal still follows the same fundamental principle: all children of a node must be visited before the node itself. However, instead of dealing with just a left and right subtree, the traversal algorithm must iterate through an entire list or array of child nodes. Each child is recursively traversed in order, and only once all children have been visited does the traversal visit the current node. This means the order becomes: first child, second child, third child, ..., nth child, then root. This is commonly used in file system trees or abstract syntax trees, where a node might have an arbitrary number of children. The key idea remains unchanged—processing all subcomponents before the parent node. The recursive implementation is easily adaptable by looping through each child rather than hardcoding left and right, making the algorithm flexible and scalable for non-binary structures.

Yes, post-order traversal can be implemented iteratively using a single stack by simulating the recursive behaviour through careful tracking of visited nodes. This method is more complex than the two-stack approach but uses less memory. The key idea is to maintain a pointer to the previously visited node and to examine the current node at the top of the stack. If the current node has no children or its children have already been processed, it is safe to visit the node. Otherwise, push the right and then the left child (if they exist) onto the stack. The algorithm checks whether to backtrack or continue descending, and it maintains a reference to determine whether the right subtree has already been visited. This approach avoids recursion entirely and is useful in systems where the call stack depth is limited or where recursion could cause a stack overflow. It requires more control logic but offers increased space efficiency.

Post-order traversal can be inefficient in scenarios where early access to root nodes is beneficial or required. For example, in search operations where the goal is to locate a node with a specific value, post-order traversal will only reach the root node after visiting all children, meaning it may take significantly longer to find the desired node compared to pre-order or in-order traversal. Similarly, in tree-building or structure-generation tasks, where parent nodes must be constructed before or alongside their children, post-order traversal’s deferred root visitation is counterproductive. It is also inefficient in shallow trees with many nodes, as the traversal processes every subtree completely before making progress toward the root, resulting in redundant work when only partial tree access is necessary. Furthermore, post-order’s recursive implementation risks causing stack overflows with deeply nested trees, especially when tail call optimisation is unavailable. In such cases, iterative traversal or alternative methods may be more efficient.

In threaded binary trees, which include extra pointers to help traverse the tree without a stack or recursion, adapting post-order traversal is challenging because the nature of the traversal requires visiting a node only after all its descendants. Unlike in-order or pre-order traversal, post-order does not have a clear “next” node until all children are processed. To support post-order traversal in such structures, additional state tracking is usually needed, such as using markers to track visited nodes or incorporating temporary flags within the tree nodes. Without parent pointers, the traversal cannot backtrack directly to a parent, which complicates determining when to visit a node after processing its children. Therefore, most post-order implementations in such cases still resort to using an auxiliary stack or modify the tree structure temporarily. Some advanced threaded tree designs include reverse threads or dedicated post-order threading mechanisms, but these are more complex and less commonly implemented due to the inherent difficulty of post-order sequencing without full backtracking support.