What type of tree twigs are safe for me to
collect for Obi to chew on?

Do a google search for bird-safe trees. You'll find several lists of trees that are considered safe and those that are considered unsafe. There will probably be several trees on the safe list that are available in your area, and you can collect branches from those.

I did that earlier but got confused in the end. Some trees are listed as safe on one list and are considered dangerous on other lists.

Just to be on the safe side of things I only used branches of trees I couldn't find on the 'nono' list anywhere..

When the same plant can be found on both the safe list and the dangerous list, it's an indication that it is generally safe but may cause problems for sensitive individuals. Some lists put a plant on the dangerous list if it can cause the slightest problem for a small minority. Others will put it on the safe list if it rarely causes problems.

One good way to learn more about these plants is to google the scientific name of the plant along with the word toxicity. This will usually turn up some sites that tell you exactly what the risks are. If the biggest problem is that it causes minor skin irritation for a very small number of people, you probably don't need to worry about it.

Thank you.... Apple and willow seem to be the safest
ones to use in this area...