Traversing N-ary Trees: DFS and BFS Extensions

Welcome back! In our last lesson, we built a robust iterative solution for the "Path Sum II" problem, using parallel stacks to manage the path and sum states during a preorder traversal. This demonstrated how a standard traversal algorithm can be adapted to solve more complex problems by carrying additional state.

Today, we'll generalize our traversal knowledge. All the trees we've worked with so far have been binary trees, where each node has at most two children. We will now extend our iterative DFS and BFS templates to handle N-ary trees, where a node can have any number of children. This is a crucial step that demonstrates the true flexibility of these traversal algorithms.

By the end of this lesson, you will be able to adapt your iterative BFS (level order), preorder, and postorder traversal patterns from binary trees to N-ary trees, a common variation encountered in coding interviews.

From Binary to N-ary Trees

An N-ary tree is a natural extension of a binary tree. Instead of having distinct left and right child pointers, a node in an N-ary tree typically holds a list or vector of its children. This seemingly small change requires us to rethink how we add children to our stack or queue during traversal.

The fundamental principles of BFS and DFS remain unchanged. BFS still explores layer by layer, and DFS still goes deep before exploring wide. The only modification we need to make is in the step where we "visit the children."

Adapting Level Order Traversal (BFS)

Let's start with level order traversal, as its adaptation is the most straightforward. In a binary tree, for each node we dequeue, we enqueue its left child and then its right child. For an N-ary tree, the logic is almost identical: for each node we dequeue, we iterate through its list of children and enqueue them in order.

The excellent resource from Algo.monster provides a very clear, in-depth explanation of this process. It covers the intuition, the step-by-step algorithm, and a worked example.

The key takeaway is that the core BFS template you already know—using a queue and processing nodes level-by-level based on the queue's size at the start of the level—works perfectly. The only change is replacing queue.add(node.left) and queue.add(node.right) with a loop: for (Node child : node.children) { queue.add(child); }.

The time complexity is $O(N)$ because we visit each node and edge once. The space complexity is also $O(N)$ in the worst case, as the queue could hold all nodes from the widest level of the tree.

Adapting Preorder Traversal (DFS)

Next, let's adapt our iterative preorder traversal. Recall the binary tree pattern:

1. Push the root onto a stack.
2. While the stack is not empty:
   1. Pop a node.
   2. Process node.
   3. Push node.right (if it exists).
   4. Push node.left (if it exists).

We push the right child before the left child because the stack is a Last-In-First-Out (LIFO) structure. This ensures we process the left subtree before the right subtree.

To extend this to N-ary trees, we apply the same logic. To process children from left to right, we must push them onto the stack from right to left.

The following video and article provide clear explanations and code for this adaptation.

The core modification is simple. Instead of pushing right then left, we loop through the children's list in reverse and push each child onto the stack.

Adapting Postorder Traversal (DFS)

Finally, we tackle iterative postorder traversal, which is the most complex of the three. For binary trees, a common and robust method uses a single stack with a (node, visited) pair to distinguish the first time we see a node from the second.

• First visit: We are traversing down the tree. We push the node back onto the stack marked as "visited" and then push its children.
• Second visit: We have finished visiting all its descendants. Now we can process the node itself.

This pattern generalizes beautifully to N-ary trees. The NeetCodeIO video on this topic gives an excellent explanation of the iterative strategy.

The algorithm is as follows:

1. Push (root, false) onto the stack.
2. While the stack is not empty:
   1. Pop (node, visited).
   2. If visited is true, it means we've processed all of its children, so we can now process node's value.
   3. If visited is false, this is our first time seeing the node. We must visit its children before processing it.
      • Push (node, true) back onto the stack.
      • Push its children onto the stack in reverse order, each paired with false. (e.g., (child_N, false), (child_N-1, false), ... (child_1, false)).

This ensures that we fully explore the subtrees of all children before we finally re-visit and process the parent node.

What About Inorder Traversal?

You might be wondering about inorder traversal. For a binary tree, "inorder" has a clear definition: Left -> Root -> Right. For an N-ary tree, there is no single, standard definition. Some might define it as "process first child's subtree, then root, then remaining children's subtrees," but this is not a common pattern you'll be asked to implement. Therefore, for N-ary trees, you should focus on mastering level order, preorder, and postorder traversals.

Conclusion

In this lesson, you've successfully generalized your iterative traversal skills from binary trees to the more generic N-ary tree structure. You've seen that the fundamental algorithms for BFS and DFS remain the same, with the primary change being how a node's children are handled.

Key Takeaways:

• An N-ary tree node stores its children in a list rather than in fixed left and right pointers.
• Level Order (BFS): To adapt, simply iterate through the children list from left to right and add each child to the queue.
• Preorder (DFS): To adapt, iterate through the children list from right to left and push each child onto the stack. This ensures the leftmost child is at the top of the stack and processed first.
• Postorder (DFS): The "visited" flag pattern extends well. On the first encounter with a node, push it back marked as "visited," then push its children in reverse order. Process the node on its second encounter.

You now have a complete and flexible set of iterative traversal templates applicable to almost any tree structure you might face.

In our next lesson, we will move away from traversal problems and introduce a new and powerful data structure: the Heap, also known as a Priority Queue. Heaps are not used for visiting every node, but for efficiently managing and retrieving elements based on their priority, opening up solutions to problems like finding the "k-th largest" element, calculating a running median, and more.