Question

Suppose each node in a binary tree contains a value. Design a function that prints all the paths that sum to a specified value passed into the function. The path can start at an arbitrary node.

Short answer

Use a recursive helper function to explore all paths starting from each node and sum paths that match the target.

Step-by-step solution

Understand the Problem

We need to find paths in a binary tree where the sum of the values equals a specified target. A path can start at any node and does not need to extend to a leaf.

Define a Helper Function

Create a recursive helper function that processes each node. This function should consider all paths starting from that node, including paths that start partway down the branches.

Initialize Starting Parameters

Initialize the traversal of the binary tree. If using a recursive function, start by passing the root node and the target sum to the function.

Consider All Paths from Each Node

For each node, calculate the cumulative path sum starting from that node. If the sum equals the target, print or record the path.

Recursive Traversal of the Tree

For each node, recursively call the helper function on both its left and right children to explore additional paths, while updating the cumulative path sums.

Terminate When a Leaf is Reached

If a node is null (reached a leaf's child), return from the function as there are no further nodes to visit.

Key concepts

Path Sum Problem

The path sum problem is a fundamental question in binary tree algorithms. It requires us to find all possible paths within a binary tree where the sum of the node values on these paths equals a specific target value. Unlike some problems that limit paths from the root to a leaf, this problem allows paths to start at any node in the tree, making it more complex.
Using this approach allows for considerable flexibility. Paths can traverse down from any starting node and don't have to end at a leaf. This means for a given node, all sequences of connected nodes below it might form valid paths whose values add up to the target sum.

• Paths can start anywhere - this means from root, middle, or leaf nodes.
• They must sum to a predefined target value.

Identifying all such paths involves exploring combinations and is often paired with recursive solutions due to the nonlinear tree structures involved.

Recursive Tree Traversal

Recursive tree traversal is a natural fit for solving the path sum problem in binary trees. The recursive approach breaks down the problem into smaller subproblems, making it easier to manage the intricate paths within the tree.
By using recursion, we can effectively explore all potential paths starting from any node. Here’s how recursion works in this context:

• At each node, initiate a check if this node starts a valid path whose sum can reach the target.
• Recursively apply the analysis on both the left and right subtrees of the node, updating variables that track the cumulative sum.

This method is efficient as it leverages the inherent structure of trees, calling smaller versions of the same task repeatedly until leaf nodes are reached, allowing for a full exploration of paths beginning at every node.

Cumulative Path Sum

Cumulative path sum is crucial when checking if there is a path whose sum of node values matches the target. It involves keeping a running tally of the sum of values along each path explored from the starting node.
When using recursion:

• Begin accumulation at the starting node with an initial value of zero. Incrementally add each visited node's value as the path traverses deeper.
• For each node and path segment, compare the cumulative sum to the target. If a match is found, record the path as valid.

Maintaining an accurate cumulative sum requires careful handling, especially as the path backtracks during recursion. This ensures previous sums don't incorrectly influence new paths being explored.

Tree Data Structure Analysis

Analyzing the data structure of a tree is essential to solving any tree-related problem, including the path sum problem. A tree is a hierarchical structure where each node might have children nodes.
For the path sum problem, understanding the tree structure helps in efficient traversal, allowing for effective algorithm design.

• Binary trees are structured such that each node has at most two children, which is a key structural constraint.
• Knowing the tree structure allows us to implement functions correctly, as traversal patterns depend heavily on whether we go left or right in the tree.

Tree analysis also involves examining the properties of nodes and how these are manipulated or processed in the algorithm. This analysis aids in optimizing the path-finding search, ensuring that we account for all viable paths and use computational resources effectively.