Binary Tree Paths

To solve the problem of finding all root-to-leaf paths in a binary tree, we can use a depth-first search (DFS) approach. We can start from the root of the tree and recursively visit all nodes, keeping track of the current path. Once a leaf node is reached, we add the path to the result.

Here’s the code to achieve this:

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class Solution:
    def binaryTreePaths(self, root: Optional[TreeNode]) -> List[str]:
        # If the tree is empty, return an empty list
        if not root:
            return []

        # Function to perform DFS and collect paths
        def dfs(node, current_path, paths):
            if not node:
                return

            # Add the current node's value to the current path
            current_path += str(node.val)

            # If the current node is a leaf, add the current path to the paths list
            if not node.left and not node.right:
                paths.append(current_path)
                return

            # If the current node has children, add '->' to the current path and continue DFS
            current_path += "->"
            dfs(node.left, current_path, paths)
            dfs(node.right, current_path, paths)

        # Initialize an empty list to store the paths
        paths = []

        # Start DFS from the root
        dfs(root, "", paths)

        return paths

The dfs function recursively visits all nodes and constructs the path from the root to each leaf node. The current_path variable keeps track of the current path as a string, and the paths list stores all the root-to-leaf paths.

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# Definition for a binary tree node.
# class TreeNode
#     attr_accessor :val, :left, :right
#     def initialize(val = 0, left = nil, right = nil)
#         @val = val
#         @left = left
#         @right = right
#     end
# end
# @param {TreeNode} root
# @return {String[]}

def construct_paths(root, path, paths)
    if root
        path += root.val.to_s

        if root.left.nil? && root.right.nil?
            paths << path
        else
            path += '->'
            construct_paths(root.left, path, paths)
            construct_paths(root.right, path, paths)
        end
    end 
end

def binary_tree_paths(root)
    paths = []
    construct_paths(root, '', paths)
    return paths
end

Identifying Problem Isomorphism

“Binary Tree Paths” can be approximately mapped to “Path Sum II”.

Both problems deal with finding paths in a binary tree, but they have different objectives. In “Binary Tree Paths”, you’re asked to find all root-to-leaf paths and return them as a list of string representations. In “Path Sum II”, you need to find all root-to-leaf paths where each path’s sum equals a target sum, returning these paths as a list of integer arrays.

Despite the difference in the final output, both problems involve traversing a binary tree from the root to each leaf node and maintaining a record of the path taken. Hence, the main skills being tested are tree traversal and path recording.

“Path Sum II” is more complex as it introduces the concept of a target sum, which requires additional checking and calculation at each step.

“Binary Tree Paths” requires knowledge of binary trees, depth-first search (DFS), and how to manipulate and generate strings. Here are 10 problems as preparation:

  1. 100. Same Tree: This problem can help you practice how to traverse a tree using DFS.

  2. 101. Symmetric Tree: You can practice DFS while comparing two different branches of a tree.

  3. 104. Maximum Depth of Binary Tree: This will give you a good understanding of DFS in a tree.

  4. 107. Binary Tree Level Order Traversal II: Practicing breadth-first search (BFS) can help you understand the difference between DFS and BFS, which can be useful for binary tree problems.

  5. 108. Convert Sorted Array to Binary Search Tree: This problem will give you a better understanding of the structure of binary trees.

  6. 110. Balanced Binary Tree: Understanding what a balanced binary tree looks like can be useful for solving binary tree problems.

  7. 111. Minimum Depth of Binary Tree: This is another problem that helps practice tree traversal.

  8. 112. Path Sum: This problem is similar to “Binary Tree Paths” but only asks whether a path exists, rather than requiring you to return all paths.

  9. 257. Binary Tree Paths: Although not a lesser complexity problem, solving this would directly help you understand the problem at hand.

  10. 226. Invert Binary Tree: Inverting a binary tree is a good exercise for understanding tree traversal and manipulation.

These cover binary trees, perform depth-first searches, and handle strings.

Clarification Questions

What are the clarification questions we can ask about this problem?

Identifying Problem Isomorphism

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# Definition for a binary tree node.
# class TreeNode:
#     def __init__(self, val=0, left=None, right=None):
#         self.val = val
#         self.left = left
#         self.right = right


class Solution:
    def binaryTreePaths(self, root: TreeNode) -> List[str]:
        result = []

        def helper(node, cur):
            if not node:
                return

            if not node.left and not node.right:
                result.append( cur + [str(node.val)] )
            else:
                helper(node.left, cur + [str(node.val)] )
                helper(node.right, cur + [str(node.val)] )

        helper(node=root, cur=[])
        return [ *map('->'.join, result) ]

Problem Classification

This problem can be classified based on the terminology used in the problem statement as follows:

  • Problem Domain: Binary Trees, Path Finding.
  • Techniques Used: Tree Traversal (Depth-First Search or Breadth-First Search may be implied).
  • Concepts Involved: Root, Leaf, Path (from root to leaf).
  • Data Structures: Trees, Paths (often represented as arrays or lists of nodes).

This problem falls under the category of tree traversal where the main goal is to identify all paths from the root of a binary tree to its leaves. This involves understanding the concept of root, leaf, and path in the context of a binary tree, which is a critical part of the problem domain. Depending on the specifics of the problem, techniques used could include different types of tree traversal such as depth-first search (DFS) or breadth-first search (BFS). Finally, the problem uses trees as the main data structure, and paths are often represented as arrays or lists of nodes.

Language Agnostic Coding Drills

  1. Understanding Tree Nodes: The first step is to understand the structure of a tree node. In this case, each node contains a value, and two pointers, one for the left child and one for the right child.

  2. Basic Tree Traversal: Get comfortable with the basics of tree traversal. In this problem, a depth-first approach is taken where we traverse down each branch as far as possible before backtracking.

  3. Path Tracking: In the traversal process, we need to keep track of the path from the root to the current node. In the helper function, cur is the variable used for this purpose, which stores the path from the root to the current node.

  4. Identifying Leaf Nodes: A key aspect of this problem is identifying leaf nodes (nodes with no children). This is done with the check if not node.left and not node.right.

  5. Adding Paths to Result: Once a leaf node is reached, the path from the root to this leaf node (which is stored in cur) is added to the final result.

  6. Recursive Function: A helper function is used to perform the traversal, path tracking, leaf node identification, and result storage. This function is recursively called on the left and right children of each node, with the current path updated accordingly.

  7. Formatting Paths: Finally, the paths, which are stored as lists of strings, are joined with ‘->’ to match the desired output format.

To create a solution, you would follow these steps:

  • Create a helper function to perform the depth-first traversal and handle the path tracking and result storage.
  • Call this helper function on the root of the tree to start the process.
  • Once all paths have been found, format them correctly.
  • Return the result.

This problem is an application of tree traversal techniques, and the associated concepts of path tracking, leaf node identification, and result storage. It demonstrates how a seemingly complex problem can be broken down into smaller, manageable tasks.

Targeted Drills in Python

  1. Understanding Tree Nodes: We start by understanding and implementing a binary tree node.

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    class TreeNode:
    def __init__(self, val=0, left=None, right=None):
        self.val = val
        self.left = left
        self.right = right

  2. Basic Tree Traversal: The next step is practicing tree traversal. Here’s a simple recursive depth-first traversal which prints the value of each node.

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    def print_values(node):
    if not node:
        return
    print(node.val)
    print_values(node.left)
    print_values(node.right)

  3. Path Tracking: Now, let’s track the path from root to current node while traversing.

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    def print_paths(node, cur=[]):
    if not node:
        return
    cur.append(str(node.val))
    print("->".join(cur))
    print_paths(node.left, cur)
    print_paths(node.right, cur)
    cur.pop()

  4. Identifying Leaf Nodes: Let’s modify the above function to only print the paths when a leaf node is reached.

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    def print_leaf_paths(node, cur=[]):
    if not node:
        return
    cur.append(str(node.val))
    if not node.left and not node.right:
        print("->".join(cur))
    else:
        print_leaf_paths(node.left, cur)
        print_leaf_paths(node.right, cur)
    cur.pop()

  5. Adding Paths to Result: Now, instead of printing the paths, we store them in a list.

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    def get_leaf_paths(node, cur=[], result=[]):
    if not node:
        return result
    cur.append(str(node.val))
    if not node.left and not node.right:
        result.append("->".join(cur))
    else:
        get_leaf_paths(node.left, cur, result)
        get_leaf_paths(node.right, cur, result)
    cur.pop()
    return result

Finally, these concepts can be combined and slightly adjusted to solve the problem in the question.

You’ll need to create a new list for cur and result each time you call the final function, as Python uses the same list across different calls to the function if it’s used as a default parameter.