Before putting any plants in your aviary, you will want to check if they are toxic to birds.
Flag iris is listed as a toxic plant - the leaves, stem, and bulb contain a toxin called irisine that causes GI distress, nausea, vomiting and diarrhea. It can also cause skin irritation and dermatitis. I don't recommend it as an aviary plant.
Ivy is also potentially toxic. Specifically, English ivy, also known as California or sweetheart ivy, contains a triterpanoid saponin which causes irritation of the mouth and gastrointestinal tract when consumed.
Most true palms are bird-safe. But there are some popular house plants that are commonly referred to as palms, but are not true palms, just look-alikes. Some of these "palm-like" plants are not safe for birds. Before moving the tree into the aviary, you will want to confirm what type of palm tree it is so you can determine if it would be an appropriate choice for your aviary.