Can I set up a Sukka under my backyard tree, and use its branches for the schach?
No. A tree, connected to the ground, is unfit for Schach, or Sukka roof. (Sukka _walls_ can be made out of any sturdy material.) Valid candidates for Schach are derived by the Talmud (Sukka 12a) from the Torah’s statement (Deuteronomy 16:13) “Celebrate seven days of Sukkos by gathering the remnants of your wheat and wine production,” meaning straw and harvested grape vines. Thus a. schach must have grown from the ground, (excluding plaster, metal or leather), b. it must already have been harvested, i.e. not attached to the ground, (excluding live tree branches), c. it must not be a utensil, such as wooden bowls, and d. it cannot be edible, such as lettuce. (See Shulchan Aruch 629 for details on kosher Schach requirements). Jewish Sukka stores often sell bamboo poles, or mats of wooden slats, which are conveniently rolled onto the top of the Sukka, and rolled up and put away after Sukkos is over. One cannot build one’s Sukka with valid Schach underneath a tree, either, or underneath a roof, for the shade of the tree renders the kosher Schach’s shade moot. One should also not use the tree as Schach, and then sever the branches from the trunk so that they qualify as ‘harvested’. This would violate the principle of ‘build — not from the built’, meaning that Schach must be laid onto the Sukka in its kosher form, and not laid first and corrected later (Shulchan Aruch O.C. 626:2).