Path In Zigzag Labelled Binary Tree

In an infinite binary tree, rows are labeled alternately:

• Odd-numbered rows (1st, 3rd, …) are labeled left → right.

• Even-numbered rows (2nd, 4th, …) are labeled right → left.

Given a node label, return the path of labels from the root (1) to that node.

Idea:
Let row index be 0-based (row 0 has label range [1,1], row 1 is [2,3], row 2 is [4,7], …).
For a node at row r with zigzag label label:

1. Convert to its normal (left→right) index within the row:

   start = 2^r, end = 2^(r+1) - 1
   idx = label               if r is even
       = start + end - label if r is odd
   

2. Parent’s normal index is idx // 2, which lies on row r-1.

3. Convert that parent normal index back to the zigzag label of row r-1:

   prevStart = 2^(r-1), prevEnd = 2^r - 1
   parent_label = parent_index                  if (r-1) is even
                = prevStart + prevEnd - parent_index  if (r-1) is odd
   

Repeat until reaching label 1. Reverse the collected list for root→node order.