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Databricks·Software Engineer·Technical Phone Screen·Senior

Senior
Jun 2026

Summary

Databricks coding interview with a pretty gnarly tree problem that requires you to navigate a recursively defined structure without ever building it. The math is elegant once it clicks but getting there under pressure is a different story.

Questions Asked (1)

Q1

Given a k-order Fibonacci tree where nodes are labeled in preorder from 0 to size(k)-1, and the tree is defined recursively with a (k-2)-order left subtree and a (k-1)-order right subtree, find the path between two node ids a and b without constructing the tree in memory.

Algorithms & Data StructuresTechnical Trade-offs
Author's notes

This one took me a while to even understand what was being asked.

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AI HintsAI Generated

Suggested Approach

Use the recursive structure to compute the preorder intervals of subtrees and determine the lowest common ancestor (LCA) without building the tree. Then, derive the path by recursively finding the path from the LCA to each node, and concatenate them appropriately.

Pro tip: Emphasize that the preorder labeling allows O(k) time and O(1) space per recursive step, and that the approach generalizes to any recursively defined tree with known subtree sizes.

1. Understand the tree structure and labeling

Recognize that the tree is defined recursively: a k-order tree has a (k-2)-order left subtree and a (k-1)-order right subtree, and nodes are labeled in preorder starting from 0. The size of a k-order tree follows a Fibonacci-like recurrence: size(k) = size(k-2) + size(k-1) + 1, with base cases size(0)=1, size(1)=1 (or similar).

2. Compute subtree sizes and intervals

Precompute or compute on the fly the size of each k-order tree. For a given node id, determine which subtree it belongs to by comparing the id with the root (id 0), the left subtree interval [1, size(k-2)], and the right subtree interval [1+size(k-2), size(k)-1].

3. Find the lowest common ancestor (LCA)

Recursively determine the LCA of nodes a and b by checking if they fall into the same subtree. If they are in different subtrees or one is the root, the current root is the LCA. Otherwise, recurse into the appropriate subtree with adjusted node ids.

4. Construct the path from LCA to each node

For each node, recursively find the path from the LCA to that node by traversing down the tree, recording the sequence of node ids. Since the tree is not built, use the same interval logic to decide whether to go left or right.

5. Combine paths to form the full path

The full path from a to b is the path from a up to the LCA (reversed) followed by the path from the LCA down to b. Ensure no duplicate LCA node.

Key Points to Mention

  • Fibonacci recurrence for subtree sizes: size(k) = size(k-2) + size(k-1) + 1
  • Preorder labeling: root at 0, left subtree occupies next size(k-2) nodes, right subtree occupies remaining size(k-1) nodes
  • Recursive traversal without explicit tree construction, using only node ids and subtree sizes
  • Time complexity: O(k) for finding LCA and paths, where k is the order of the tree
  • Space complexity: O(k) for recursion stack, or O(1) if iterative
  • Handling base cases: k=0 or k=1 trees are single nodes

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.