Path In Zigzag Labelled Binary Tree

Path In Zigzag Labelled Binary Tree

What is this problem about?

The Path In Zigzag Labelled Binary Tree problem gives you a "zigzag" labelled binary tree where odd levels are labelled left-to-right and even levels are labelled right-to-left. Given a node label, find the path from root to that node. This coding problem uses math to navigate the labelling pattern. The math, binary tree, and tree interview pattern is demonstrated.

Why is this asked in interviews?

Bloomberg asks this to test mathematical reasoning about binary tree structure with non-standard labelling. The key: convert zigzag label to normal binary tree label by reflecting within the level, then trace the path using parent relationships.

Algorithmic pattern used

Reflection and parent tracing. For a node at level h (root is level 1), its level has nodes labelled 2^(h-1) to 2^h-1. The zigzag label label in an even level corresponds to normal label 2^h - 1 - (label - 2^(h-1)) = 2^h + 2^(h-1) - 1 - label. Find normal label, then trace parents: parent = label // 2 for normal tree. At each step, reflect back to get the zigzag path label.

Example explanation

label=14. Level h=4 (2^3=8 ≤ 14 < 16=2^4). Level 4 is even (right-to-left). Normal label of 14: 2^4 + 2^3 - 1 - 14 = 16+8-1-14 = 9. Parent of 9 = 4. Level 3 is odd: 4 stays as 4. Reflect: 2^3+2^2-1-4=11. Parent of 4=2. Level 2 even: reflect: 2^2+2-1-2=3. Parent of 2=1. Level 1: root=1. Path: [1,3,11,14].

Common mistakes candidates make

• Not reflecting the label at each level for the output path.
• Incorrect level computation (level = floor(log2(label)) + 1).
• Reflecting at wrong levels (only reflect at levels where labelling is reversed).
• Off-by-one in level boundaries.

Interview preparation tip

Zigzag-labelled tree problems require understanding the mapping between zigzag labels and standard binary tree positions. The reflection formula 2^h + 2^(h-1) - 1 - label converts between the two labellings at level h. Practice deriving this formula from small examples. The parent tracing in standard binary tree (label//2) combined with reflection gives the full path. Math on trees requires careful boundary handling.