The information that I have read about tree twigs is to limit them to multiseeded fruit trees (apples, pears) and avoid single seeded fruit (peach, apricot, plum). There are other good trees too, including ash, birch and a couple of others. Here is what the HRS website says:
"Fruit tree branches, such as, apricot and peach are toxic while attached to the tree but not after they’re cut and dried (a month or more). Lisa McSherry and Rusty Fayter, who package The Busy Bunny baskets, share this research. Another tip they offer for your bunny’s safety is to keep your purchases of imported baskets limited to willow, the only basket material not sprayed with pesticide."
Here is what Medirabbit has to say on the subject:
"Trees carrying drupes – single seeded fruits (including plum, cherry, apricot, and peach are toxic to rabbits.
Safe Trees:
Willow, spruce, ash, birch, maple, juniper, poplar, apple, pear, hazel and hawthorne"