Learn and practice the Tree Depth-First Search, Tree Breadth-First Search, and Backtracking patterns to solve coding interview problems.

Grokking the Blind 75 Problems in Go

In this module, we’ll learn to work with the tree data structure and explore the use of the backtracking algorithm. Trees play a pivotal role in organizing hierarchical data, optimizing searches, and solving complex problems. We’ll learn to choose the appropriate tree for a given problem and apply the more appropriate of the two major traversal techniques, breadth-first or depth-first, to solve it. We use backtracking when seeking optimal solutions in a scenario where no possibility may be left unexplored. We’ll gain experience with using tree depth-first search to implement backtracking when the solution space is represented as a tree, and we’ll need to exhaustively explore all possible solutions. We’ll also learn to implement backtracking to solve problems where the solution space is represented using other data structures.

Exploring the Solution Space

# Statement
Given the root of a binary tree, return the maximum sum of any non-empty path.

A path in a binary tree is defined as follows:

* A sequence of nodes in which each pair of adjacent nodes must have an edge connecting them. 
    * A node can only be included in a path once at most. 
    * Including the root in the path is not compulsory.

You can calculate the path sum by adding up all node values in the path. To solve this problem, calculate the maximum path sum given the root of a binary tree so that there won’t be any greater path than it in the tree.

**Constraints:**
* $1\leq$ Number of nodes in the tree $\leq500$.
* $-1000 \leq$ `Node.value` $\leq 1000$
# Examples


# Understand the problem
Let’s take a moment to make sure you’ve correctly understood the problem. The quiz below helps you check if you’re solving the correct problem:

What is the maximum path sum for the given binary tree?
```text
   -8
   / \
  2   17
 / \  / \
1  4 24  5
```
level order = [-8, 2, 17, 1, 4, 24, 5]

What is the maximum path sum for the given binary tree?
<pre><code class="language-text"> -8
 / \
 2 17
 / \ / \
1 4 24 5
</code></pre>
level order = [-8, 2, 17, 1, 4, 24, 5]

What is the maximum path sum for the given binary tree?
```text
     9
    / \
   4   5
  / \  
-1  -2 
```
level order = [9, 4, 5, -1, -2]

What is the maximum path sum for the given binary tree?
<pre><code class="language-text"> 9
 / \
 4 5
 / \ 
-1 -2 
</code></pre>
level order = [9, 4, 5, -1, -2]

What is the maximum path sum for the given binary tree?
```text
    10
    / \
   11  12  
```
level order = [10, 11, 12]

What is the maximum path sum for the given binary tree?
<pre><code class="language-text"> 10
 / \
 11 12 
</code></pre>
level order = [10, 11, 12]

Binary Tree Maximum Path Sum

Binary Tree Maximum Path Sum

# Figure it out!
Here’s a game for you to play. Rearrange the logical building blocks to develop a clearer understanding of how to solve this problem.

# Try it yourself
Implement your solution in `main.go` in the following coding playground.

package main

type BinaryTreeNode struct {
	data  int
	left  *BinaryTreeNode
	right *BinaryTreeNode

	// below data members used only for some of the problems
    next *BinaryTreeNode
	parent *BinaryTreeNode
	count int
}

// NewBinaryTreeNode initializes and returns the BinaryTreeNode 
// type object
func NewBinaryTreeNode(data int) *BinaryTreeNode {
	t := new(BinaryTreeNode)
	t.data = data
	t.left = nil
	t.right = nil
	t.next = nil
	t.parent = nil
	t.count = 0
	return t
}

# Quick Fix\nYou are Edward, a Senior Software Engineer who specializes in coding interview patterns, data structures and algorithms, and is great at debugging issues in code.\n\n- Edward, please keep in mind that you do not have to print the data of the placeholders. Also, do not mention anything about yourself.\n\nEdward, you are given the following data in the placeholders:\n- Function name: `maxPathSum`\n- Candidate's submitted solution code: USER_CODE\n- Error: ERROR_DESCRIPTION\n- Failed test case: FAILED_TEST_CASE\n- Coding language: `Go`\n\n- Edward, your task is to understand the type of error that is occurring in the coding language provided.\n- Once you've understood the error as per the error’s description, and why the test case failed, you will suggest how the candidate can improve their solution code to fix the error.\n- Edward, you will not give the corrected code, just a hint on what the error tells, and how to fix it.\n- Edward, please limit your response strictly to 100 words. No more words than the limit.

# Review Code\nEdward, your current task is to review the `maxPathSum` function(s) and the other function(s) invoked in it/them.\n\n-Take your time and review the candidate's solution code.\n- Do not review any extra/helper files or functions not invoked in the solution function `maxPathSum`.\n- Do not review the code of data structure objects instantiated like a binary tree, linked list, trie, union find, etc., in the solution functions. Just review the solution/algorithm function of the problem. \n- Also, if the solution code doesn’t use the coding pattern stated above, then please encourage the candidate to reattempt the problem using that pattern corectly.

# Review time complexity\n- Please provide a brief time complexity analysis of the function(s) `maxPathSum`. \n- While calculating time complexity, include the time complexity of built-in Go functions invoked by the code. \n- The answer should be to the point in approximately 60 words.

# Review space complexity\n- Please provide a brief space complexity analysis of the function(s) `maxPathSum`. \n- While calculating space complexity, include the space complexity of built-in Go functions invoked by the code. \n- Do not include the space used for the input parameters and the space used for just returning an output (not used as a scratch space). \n- The answer should be to the point in approximately 60 words.

def evaluate_results(self) -> object:
 """
 self.__inputs is the list of inputs that are passed as input to function
 self.__actual_output is a list of outputs that is produced by the learner's code
 return : self.__edu__result object
 """
 
 from collections import deque
 

 def create_binary_tree(level_order):
 if not level_order:
 return None
 
 root = TreeNode(level_order[0])
 q = deque([root])
 i = 1
 while i < len(level_order) and q:
 curr = q.popleft()
 
 if level_order[i] != None:
 curr.left = TreeNode(level_order[i])
 q.append(curr.left)
 i += 1
 
 if i < len(level_order) and level_order[i] != None:
 curr.right = TreeNode(level_order[i])
 q.append(curr.right)
 i += 1
 
 return root
 
 

 # multiple outputs will be added at each index respectively for a single test_case
 expected_outputs = [] 
 inp = self.__inputs[0]
 inp_root = create_binary_tree(inp)
 expected = max_path_sum(inp_root)
 expected_outputs.append(expected)
 actual = self.__actual_outputs[0]
 
 if json.dumps(expected) == json.dumps(actual):
 self.__edu_result.store_result(expected_outputs, TestCaseResult.PASS)
 else:
 self.__edu_result.store_result(expected_outputs, TestCaseResult.FAIL)

 return self.__edu_result


def max_contrib(root):
 if not root:
 return 0

 max_left = max_contrib(root.left)
 max_right = max_contrib(root.right)

 left_subtree = 0
 right_subtree = 0

 if max_left > 0:
 left_subtree = max_left
 if max_right > 0:
 right_subtree = max_right

 value_new_path = root.data + left_subtree + right_subtree

 max_sum = max_contrib.max_sum
 max_contrib.max_sum = max(max_sum, value_new_path)

 return root.data + max(left_subtree, right_subtree)

def max_path_sum(root):
 max_contrib.max_sum = float('-inf')
 max_contrib(root)
 return max_contrib.max_sum


class TreeNode:
 def __init__(self, data):
 self.data = data
 self.left = None
 self.right = None

// Definition of a binary tree node 
// type TreeNode[T any] struct {
// 	Data  T
// 	Left  *TreeNode[T]
// 	Right *TreeNode[T]
// }

package main
import (. "golang-test-code/ds_v1/BinaryTree") // TreeNode[T]

func maxPathSum(root *TreeNode[int]) int {
	
	// Replace this placeholder return statement with your code
	return -1
}

binary_tree_maximum_path_sum (1).yaml

testcases:
  - name: "testcase 1"
    inputs:
    - 1 : [-8, 2, 17, 1, 4, 19, 5]
    output: 
    - 1 : 41
  - name: "testcase 2"
    inputs:
    - 1 : [7, 3, 4, -1, -3]
    output: 
    - 1 : 14
  - name: "testcase 3"
    inputs:
    - 1 : [-10, 9, 20, 12, 11, 15, 7]
    output: 
    - 1 : 46
  - name: "testcase 4"
    inputs:
    - 1 : [1, null, 2, null, 3, null, 4, null, 5]
    output: 
    - 1 : 15
  - name: "testcase 5"
    inputs:
    - 1 : [0]
    output: 
    - 1 : 0

def sum(input_1, input_2):
    return input_1+input_2

def validate_inputs(self) -> None:
 """
 self.__inputs is the list of inputs that are passed as input to function
 store the result using the self.store_result(error_msg, status)
 return : void
 """
 
 from collections import deque

 def level_order_traversal(root):
 if not root:
 return []
 
 result = []
 q = deque([root])
 while q:
 curr = q.popleft()
 
 if curr:
 result.append(curr.data)
 q.append(curr.left)
 q.append(curr.right)
 else:
 result.append(None)

 if all(val == None for val in q):
 break 
 
 return result
 
 def is_valid_level_order(level_order):
 if not level_order:
 return True # An empty list is a valid level-order traversal

 n = len(level_order)
 i = 0
 root = TreeNode(level_order[i])
 q = deque([root])
 i = 1

 while i < n and q:
 curr = q.popleft()

 if level_order[i] is not None:
 curr.left = TreeNode(level_order[i])
 q.append(curr.left)
 i += 1

 if i < n and level_order[i] is not None:
 curr.right = TreeNode(level_order[i])
 q.append(curr.right)
 i += 1
 
 # Remove None values after the last integer
 while level_order and level_order[-1] is None:
 level_order.pop()
 
 return level_order_traversal(root) == level_order

 # inputs can be accessed using self.__inputs starting from 0 to len(self.__inputs) - 1
 input_list = self.__inputs[0]
 # inputs can be accessed using self.__inputs starting from 0 to len(self.__inputs) - 1
 if not all(-1000 <= num <= 1000 for num in input_list if num is not None):
 self.store_result ("'Node' must consist of values from -1000 to 1000 only", ValidationFunction.FAIL)
 elif (len(input_list) != 0 and input_list[0] is None):
 self.store_result ("Not a valid value of type TreeNode", ValidationFunction.FAIL)
 elif not (1 <= sum(1 for value in input_list if value is not None) <= 500):
 self.store_result ("Total nodes in the tree are expected to be between 1 and 500", ValidationFunction.FAIL)
 elif not is_valid_level_order(input_list):
 self.store_result ("Not a valid value of type TreeNode", ValidationFunction.FAIL)

 else:
 self.store_result("", ValidationFunction.PASS)


class TreeNode:
 def __init__(self, data):
 self.data = data
 self.left = None
 self.right = None

Try to solve the Binary Tree Maximum Path Sum problem.

Tree Breadth-first Search

Tree Depth-first Search

Backtracking

Conclusion

Binary Tree Maximum Path Sum

Statement

Examples

Understand the problem

Figure it out!

Try it yourself