Bring your doo dad, frou frou, adornment, decoration, gimcrack, knick knack, treasure, or trinket with your geo name on it and hang it on the tree. The tree is in a less traveled location but in a nice little nook of a small courtyard. You may at certain times of the day encounter a muggle or two.
You can sign the log without bringing anything to hang. But if you bring something, it will add to the fun!