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This forum is for advice about initial treatment given to your injured/sick bird until a qualified avian veterinarian is available. THIS IS NOT MEANT TO REPLACE VET CARE
It is better to identify a type of tree from a leaf or from the bark on the tree. It is also wise to collect fallen branches after a wind storm so they are cleaner. I bought a tree identification book and identified the trees nearby where I live and found very few were in the safe list for birds. I do not recommend that you just pick up a branch because it looks nice.
Of course you do not have a parrot, you have a type of bird found in the wild. If you know of a vet or avian rehabber perhaps they could let you know if the unsafe woods for parrots and also unsafe for wild birds. However it could be your bird was never exposed to the natural woods where trees grow.
Now that the emergency is over we should not be posting here any more. You can always start a new one elsewhere.