A #Christmas #Mathematics #puzzle!

I am wrapping garland around my Christmas tree in a long continuous spiral, top to bottom.

I have two lengths of garland, equal in length. When the first length runs out, I’ll start the second one, which will finish the job.

How high up the tree does the first length of garland run out and the second length start?

For simplicity, the garland is a one-dimensional string and the tree is a cone resting on the floor. The garland is very long compared to the height of the tree and successive wrappings are equally spaced.

Edit: I've posted a geometric asymptotic solution as a reply here: https://old.mermaid.town/@futzle/115652589812829416