In my search for the everlasting, environmentally friendly label for trees, I’ve found the Dymo Rhino M1011 which embosses aluminium tape.
About 15cm of the aluminium tape is curled, so that as the branch expands, the label always fits.
The downside is that, as of December 2023, the labeller retails new for about £500, and the 4.8m aluminium tape is about £14. I bought my labeller secondhand on eBay for £120, and found a supplier on eBay selling the tape for about £6.
I have a lot of labels to make, but if you don’t have many, an alternative is to use recycled aluminium cans and a manual alphabet punch set.