Which branch decides cases a about treaties with other countries?

There are two possibilities. Usually, the President and the Secretary of State approach the other country and create the agreement. As chief executive, the President is authorized to sign the agreement with other countries without congressional approval. The Senate must be notified within 60 days of its signature.

Another way to go is for the agreed upon treaty to be transmitted to the Senate Foreign Relations Committee after the President, the other countries and the Secretary of State have given their approval. Their duty is to "advise and consent." Then, the agreement is put to a vote. 2/3 of the Senate must approve the agreement.