Problem Description

The problem provides a binary tree (where each node has at most two children) and asks you to transform it into a specific string format based on the rules of preorder traversal. This traversal visits the root, followed by the left subtree, and then the right subtree. As you navigate the tree, the goal is to create a string that represents the tree structure via parentheses. Each node's value must be captured in the string, and for non-leaf nodes, its children should be represented inside parentheses. Importantly, if a node has an empty right child, you still need to use a pair of parentheses to indicate the left child, but you are not required to include parentheses for an empty left child if there is also no right child. The challenge is to omit all unnecessary empty parentheses pairs to avoid redundancy in the resulting string.

Flowchart Walkthrough

To determine the best algorithm for LeetCode problem 606, "Construct String from Binary Tree," let's utilize the Flowchart. Here's the analysis according to the given flowchart nodes and edges:

Is it a graph?

• Yes: The binary tree is a type of graph where each node can have at most two children.

Is it a tree?

• Yes: Specifically, the problem involves a binary tree, which is a type of tree with unique paths from the root to any node.

Following the "Yes" path for a tree, we should consider using the Depth-First Search (DFS) algorithm.

Conclusion: The flowchart directs us to use DFS for problems involving a tree. This method is appropriate as we need to traverse the tree to construct the string representation. DFS is ideal for exploring all nodes in a tree to ensure each node's value and its relationship to other nodes (left and right children) are correctly represented in the output string.

Intuition

The solution is grounded in the idea of depth-first search (DFS), which explores as far as possible along each branch before backtracking. This concept is perfect for the preorder traversal required by the problem. To implement the DFS, you recursively visit nodes, starting from the root down to the leaves, while building the string representation as per the rules.

Here's an intuitive breakdown of the approach:

• If the current node is None (meaning you've reached a place without a child), return an empty string since you don't need to add anything to the string.

• If the current node is a leaf node (it has no left or right child), return the node's value as a string, since no parentheses are needed.

• If the node has a left child but no right child, you only need to consider the left subtree within parentheses immediately following the node's value. There's no need for parentheses for the nonexistent right child in this case.

• If the node has both left and right children, you must include both in the string with their respective parentheses following the node's value.

Recursive calls are made to cover the entire tree, and at each step, you construct the string according to which children the current node has. By following these recursive rules, you ensure that you only include necessary parentheses, thus preventing any empty pairs from appearing in the final output string.

Learn more about Tree, Depth-First Search and Binary Tree patterns.

Solution Approach

The solution uses a simple recursive algorithm, which is an application of the Depth-First Search (DFS) pattern. Here's how it works:

• A helper function dfs is defined that accepts a node of the tree as its parameter. This function is responsible for the recursive traversal and string construction.

• The base case of this recursion is when the dfs function encounters a None node. In binary trees, a None node signifies that you've reached beyond the leaf nodes (end of a branch). When this happens, an empty string is returned because there's nothing to add to the string representation from this node downwards.

• When the current node is a leaf (neither left nor right child exists), we simply return the string representation of the node's value because leaves do not require parentheses according to the problem rules.

• When a node has a left child but no right child, we include the left child's representation within parentheses following the node's value. However, the right side doesn't need any parentheses or representation because it's a None node.

• The most involved case is when both left and right children exist. In this scenario, we need to represent both children. Therefore, recursively the dfs function is called for both the left and the right child, and their representations are enclosed in parentheses with the format: node value(left child representation)(right child representation). This ensures that the preorder traversal order is maintained.

Additionally, the given solution employs an inner function dfs and makes use of Python's string formatting to concatenate the values and parentheses cleanly.

Here's an implementation breakdown corresponding to the code sections:

def tree2str(self, root: Optional[TreeNode]) -> str:
    def dfs(root):
        if root is None:  # Base case for NULL/None nodes.
            return ''
        if root.left is None and root.right is None:  # Leaf node case.
            return str(root.val)
        if root.right is None:  # Node with only left child case.
            return f'{root.val}({dfs(root.left)})'
        # Node with both left and right children case.
        return f'{root.val}({dfs(root.left)})({dfs(root.right)})'

    return dfs(root)  # Initial call to the dfs function with the [tree](/problems/tree_intro) root.

• At the start, dfs(root) is called with the root of the binary tree.

• The if conditions inside the dfs function handle the cases mentioned previously and are responsible for constructing the string as per the rules and format required by the problem statement.

• The resulting string is built up step-by-step through recursive calls until the entire tree has been traversed, thus providing the correct string representation.

Example Walkthrough

Let’s consider a small binary tree example to illustrate the solution approach. The binary tree used in this example is as follows:

     1
    / \
   2   3
  / 
 4 

In this tree:

• The root node (1) has two children (2 and 3).
• The left child of the root (2) has a left child of its own (4) and no right child.
• The right child of the root (3) is a leaf node and has no children.
• The leaf node (4) also has no children.

Using the problem's rules and solution approach, we perform a preorder traversal to construct the string:

1. Start with the root (1). Since it's not None and not a leaf, the string representation starts with its value. So far we have the string "1".

2. Next, visit the left child (2) of node (1). Again, as it is not a None or a leaf node and it has a left child but no right child, we include its left child wrapped in parentheses. So, we now have "1(2".

3. We now go to the left child of node (2), which is node (4). This node is a leaf, so we simply return its value as a string, resulting in the substring "4". Adding this to the previous string, we get "1(2(4)".

4. Since node (2) has no right child, we do not need to include it, and we close the parentheses for node (2). Our string is now "1(2(4))".

5. Next, we consider the right child of the root, which is node (3). It is a leaf node, so we just need its value for the string, wrapped in parentheses because its sibling node (2) has children. Our string becomes "1(2(4))(3)".

6. Finally, since we have represented both children of the root (1), we close the representation of the root. The final string is "1(2(4))(3)", with no unnecessary parentheses.

Using the given solution approach, the recursive function dfs would follow these steps internally to build the string representation smoothly. Here's the implementation of the tree structure in code and how the dfs function builds the string:

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

def tree2str(root: Optional[TreeNode]) -> str:
    def dfs(root):
        if root is None:
            return ''
        if root.left is None and root.right is None:  # Leaf node case.
            return str(root.val)
        if root.right is None:  # Node with only left child case.
            return f'{root.val}({dfs(root.left)})'
        # Node with both left and right children case.
        return f'{root.val}({dfs(root.left)})({dfs(root.right)})'

    return dfs(root)

# Example tree construction
node4 = TreeNode(4)
node2 = TreeNode(2, node4)
node3 = TreeNode(3)
root = TreeNode(1, node2, node3)
# Call to transform tree into string
print(tree2str(root))  # Output should be "1(2(4))(3)"

Following the recursive process described above using dfs, we obtain the expected string representation of the binary tree based on the problem’s guidelines.

Solution Implementation

Time and Space Complexity

The time complexity of the given code is O(n), where n is the number of nodes in the binary tree. This is because the code performs a depth-first search (DFS) and visits each node exactly once when generating the string representation of the binary tree.

The space complexity of the code is also O(n) in the worst case when the tree is highly skewed (i.e., each node has only one child). This comes from the recursive call stack that could have a maximum depth equivalent to the number of nodes in the tree if the tree degenerates into a linked list. In a balanced tree, the space complexity would be O(log n) due to the height of the balanced tree being log n.

Learn more about how to find time and space complexity quickly using problem constraints.