Live (green wood) branches are a wonderful accessory for the cage. Not only will your Joey run along them, but it will also enjoy being able to chew on them. Branches should be replaced when they become dry or brittle, or when about 1/2 of the bark has been stripped off. Make sure you only use Fresh Eucalyptus, commercially-dried Eucalyptus is dried with preservatives, which can be harmful to your glider. Some types of wood are toxic to sugar gliders and should not be used. A brief list of safe and unsafe varieties follows: this is not a complete listing.
Acacia, Apple, Aspen, Balsa, Bamboo (real bamboo, not decorative which is toxic), Beech, Blue Spruce, Butterfly Bush, Buttonwood, Carob Tree, Cholla, Citrus (orange, lime, lemon, etc.), Cork Bark (not the wood of the cork oak, just the bark), Cottonwood, Crabapple, Crape Myrtle, Douglas Fir, Eucalyptus (), Forsythia, Grapewood (woody grape vines), Hazelnut, Hybiscus, Lilac, Madrone, Magnolia, Manzanita, Maple (except Red Maple varieties), Mimosa, Mountain Ash, Mulberry, Pear, Pecan, Plane Tree, Poplar, Rose of Sharon, Sassafras, Silk Tree, Strawberry Tree, Silver Maple, Sugar Maple, Sycamore, Redbud, Ribbonwood, Willow, Yucca.
Woods that are often considered safe, but may cause dermatological reactions in some gliders; Birch, Elm, Ficus, Dogwood, Pussy Willow, Rosewood, Rubber Tree.
Trees that bear pit containing fruits are NOT safe woods to use. These include; Apricot, Avocado, Cherry, Nectarine, Peach, Plum, Almond, Black Walnut, Cedar, Pine/Fir, etc.
Other unsafe woods include (but are not remotely limited to); Box Elder, Boxwood, Cedar, Oak, Pine, Red variety Maples, Walnut, etc.