Step-By-Step Directions From a Binary Tree Node to Another

Step-By-Step Directions From a Binary Tree Node to Another

What is this problem about?

The Step-By-Step Directions From a Binary Tree Node to Another coding problem asks you to find the shortest path between two specific nodes in a binary tree. You are given the root, the startValue, and the destValue. The path should be described using directions: 'U' for moving from a child to a parent, 'L' for moving to a left child, and 'R' for moving to a right child.

Why is this asked in interviews?

This is a popular question at companies like Google and Amazon because it combines multiple tree concepts: pathfinding, Lowest Common Ancestor (LCA), and Depth-First Search (DFS). It tests your ability to navigate tree structures in both directions (up toward the root and down toward leaves) and your skill in string manipulation to construct the final path.

Algorithmic pattern used

The core pattern is Lowest Common Ancestor (LCA). The shortest path between two nodes always goes through their LCA.

1. Find the paths from the root to the startValue and the destValue using DFS.
2. Compare the two paths to find the first index where they diverge. This divergence point represents the LCA.
3. For the start node, all steps from its position up to the LCA must be 'U'.
4. For the destination node, all steps from the LCA down to its position are the 'L' and 'R' directions found in its path suffix.
5. Concatenate the 'U's and the destination suffix to get the final path.

Example explanation (use your own example)

Root = 1, 1's left = 2, 1's right = 3, 2's left = 4, 2's right = 5. Start = 5, Dest = 3.

• Path Root to 5: "LR" (Left to 2, Right to 5).
• Path Root to 3: "R" (Right to 3).
• Divergence is at the root (index 0).
• Start path suffix after LCA: "LR". Change all to "U": "UU".
• Dest path suffix after LCA: "R".
• Result: "UUR".

Common mistakes candidates make

• Inefficient traversal: Trying to perform a BFS to find the path, which is difficult because you don't have parent pointers.
• Not using LCA: Trying to find a direct path without realizing that moving through the common ancestor is the standard way to navigate trees.
• String concatenation overhead: Repeatedly prepending characters to a string, which can be $O(N^2)$; use a list and reverse/join instead.
• Memory limits: Storing too much information during the search.

Interview preparation tip

When solving Binary Tree, Tree interview pattern problems that involve paths between nodes, always think of the Lowest Common Ancestor. It's the "bridge" that connects any two nodes in a tree.