Binary Trees: A Comprehensive Guide for Coding Interviews

Nodes, Edges, and Tree Terminology

There’s an extensive set of terms used to describe properties of binary trees. Make sure you’re familiar with these, so you’re on the same page as your interviewer when they describe the problem:

• Node: A single item in a binary tree, storing a value and pointers to child nodes
• Root: The topmost node of the tree, which has no parent.
• Leaf: A node with no children; these appear at the bottom of the tree
• Edge: The connection between two nodes.
• Sibling: Nodes that share the same parent.
• Subtree: Any node of the tree along with its descendants.

Trees are often measured by their height or depth — the number of levels between the root and the furthest leaf node. Depending how it’s defined, a tree with just one node might have a height of zero or one. Ask your interviewer if you’re unsure how they’re using the term.

Types of Binary Trees

Depending on their structure, some binary trees can have special names.

• In a complete binary tree, every level is full except possibly the last level. The last level is filled left to right. Complete binary trees are the basis for heaps.
  
• In a full binary tree, every node has exactly 0 children (leaf nodes) or 2 children (internal nodes). Full binary trees often come up in parsing problems.
  
• In a perfect binary tree, no more nodes can be added without creating a new level. Perfect binary trees of height h have 2^{h + 1} - 1 nodes. Tournament brackets often resemble perfect binary trees.
  
• In a balanced binary tree, every node’s subtrees have similar heights (usually, they differ by at most one). The balanced property ensures operations like insertion, deletion, and search are .

Tree Traversal Techniques

When working with binary trees, a common operation is to visit each node exactly once to process or inspect its data. This process is known as tree traversal. Recognizing the appropriate traversal technique for a given problem is a crucial skill in coding interviews.

Because of the inherent recursive structure of trees (each node has smaller left and right subtrees), tree traversals are often implemented using recursive algorithms. Let's explore the common traversal methods.

In-order traversal

In an in-order traversal, we visit nodes in the following order:

1. Do an in-order traversal of the left subtree.
2. Visit the root node.
3. Do an in-order traversal of the right subtree.

Here’s the order we’d visit each node in our tree using an in-order traversal:

In-order traversal is particularly useful when working with Binary Search Trees (BSTs). Because of the BST property (smaller values in the left subtree, larger values in the right subtree), an in-order traversal will visit the nodes in ascending sorted order. This makes it useful for operations like:

• Printing nodes in sorted order in a BST.
• Finding the median value in a BST.
• Checking if a BST is valid.

Pre-order traversal

In a pre-order traversal, we visit nodes in this order:

1. Visit the root node.
2. Do a pre-order traversal of the left subtree.
3. Do a pre-order traversal of the right subtree.

Here’s what that order looks like:

Pre-order traversal is often used in scenarios like:

• Creating a copy of a binary tree. The root is visited first, allowing you to construct the root in the copied tree before recursively copying the subtrees.
• Tree serialization. Pre-order traversal allows you to reconstruct the tree structure when combined with a way to mark null nodes.

Post-order Traversal

In a post-order traversal, the node visiting order is:

1. Do a post-order traversal of the left subtree.
2. Do a post-order traversal of the right subtree.
3. Visit the root node.

Going back to our sample tree, here's the order when traversing this way:

Post-order traversal is frequently employed when:

• Deleting or freeing nodes in a binary tree. By processing the children first, you ensure that you don't lose access to subtree nodes when you delete the parent. This prevents memory leaks.
• Calculating directory sizes in a file system (where directories can be represented as tree nodes).

Level-order Traversal (Breadth First Search)

This one is different from the others. In a level-order traversal, the tree is explored level by level, starting from the root.

1. Visit all nodes at the current level.
2. Move to the next level and repeat.

Walking our example tree one last time:

Level-order traversal is typically implemented iteratively using a queue. It's useful for:

• Finding the shortest path between nodes in a tree.
• Printing tree nodes level by level.

Complexity Analysis

Let's analyze the time and space complexity of these traversal techniques:

For Recursive Traversal Algorithms (In-order, Pre-order, Post-order):

Time Complexity: — where N is the number of nodes in the tree. In each traversal, we visit every node exactly once and perform a constant amount of work at each node (assuming each visit operation is ).

Space Complexity: —where h is the height of the tree. The space complexity is primarily determined by the recursive call stack. In the worst case, for a skewed or unbalanced tree (e.g., a linked list-like tree), the height can be N, leading to space complexity. However, for a balanced binary tree, the height is approximately \lg(N) resulting in space complexity.

For Level-order Traversal (Breadth-First Search):

Time Complexity: — similar to the recursive traversals, we visit each node once and do work during the visit.

Space Complexity: — where W is the maximum width of the tree. In the worst-case scenario, a complete binary tree, the maximum width is approximately N/2 (at the last level). Therefore, the space complexity can be in the worst case, as the queue might hold a significant portion of the nodes at a given level. In more sparse trees, the space complexity might be less.

Binary Tree Problem-Solving Strategies

How to Recognize an Interview Problem is a Binary Tree Problem

Sometimes, the problem won't explicitly scream "binary tree!" You need to learn to spot the subtle clues that hint at a tree-based solution. Look for these indicators:

• Hierarchical Relationships: Does the problem involve parent-child relationships or branching? Think about organizational charts, file systems, or family trees – these are naturally hierarchical. If the problem describes these kinds of relationships, consider binary trees as a potential data structure.
• Recursive Structure: Binary trees are inherently recursive. If the problem can be broken down into smaller, self-similar subproblems, especially involving left and right components, it's a strong signal that a tree or recursive tree traversal is involved.
• Logarithmic Complexity: If a problem explicitly asks for an solution, that’s a hint that binary search trees might be involved. Most tree operations are , while hash-based structures often have operations and lists often have or operations.

Space Optimization Techniques

While binary tree operations often have logarithmic time complexity (assuming they’re balanced), space complexity can sometimes be a concern, especially with recursion. Here are a few techniques to consider for space optimization:

• Iterative Traversal (for recursive traversals): While recursion is often the most natural way to implement in-order, pre-order, and post-order traversals, they can be implemented using a stack. This doesn’t change the asymptotic complexity, but it does often reduce the real amount of space used.
• Morris Traversal (for in-order and pre-order): This is an advanced, constant space traversal technique for in-order and pre-order traversals. It cleverly uses the unused left and right fields of leaf nodes to create temporary links to the inorder predecessor or successor, allowing traversal without a stack or recursion. Morris traversal is an impressive technique to know, but it's very niche. Mention it to show the depth of your knowledge; few production systems actually implement it.

Prioritize clarity and correctness over premature optimization. In an interview setting, it's generally better to first implement a correct and understandable solution, even if it's not the most space-optimized. If space complexity becomes a concern or if the interviewer specifically asks about space optimization, then you can discuss or implement iterative approaches or more advanced techniques.

Common Edge Cases to Consider

Always consider these edge cases when working with trees:

• Empty Tree (Null Root): The most fundamental edge case! What should your function do if the input tree has no nodes?
• Single Node Tree: A tree with only a root node. Does your algorithm work correctly for a single node?
• Skewed Trees (Linked List-like Trees): Unbalanced trees where all nodes are on one side (either all left children or all right children). These trees can degenerate the performance of BST operations to time and space complexity. Be mindful of this worst-case scenario and consider using balanced binary trees to avoid it.
• Duplicate Values (in BSTs): If you are working with Binary Search Trees, clarify with your interviewer how duplicate values should be handled. Should they be placed in the left subtree, right subtree, or are duplicates not allowed?