What did Yosef see in the chief butler's dream that prompted him to give a favorable interpretation, bearing in mind that dreams are fulfilled according to their interpretation?
Targum Yonasan: In the first half of his dream, the three branches represented (a) the three Avos, 1 Avraham, Yitzchak and Yakov, whose children were destined to slave in cement and bricks and all kinds of hard work in the fields 2 and (b) the three redeemers (Moshe, Aharon and Miriam); 3 whereas in the second half, his squeezing out the grapes into the cup of Pharaoh represented the cup of punishment that Pharaoh would ultimately be made to drink. And the favorable interpretation was a reward for the favorable dream that he had dreamt about Yisrael. 4
Moshav Zekenim (19): A vine has roots and lasts - this shows that the butler will survive and be strengthened. The baker saw baked goods - even if one puts them in a storehouse, they spoil and deteriorate by themselves, so Yosef thought that he will not survive.
Why do the three branches represent three days, and in Pharaoh's dream, the seven sheaves represent seven years?
Refer to 40 10:1:2**.
Tosfos ha'Shalem (1, citing Moshav Zekenim): Surely the sheaves must represent years. There cannot be seven days of famine and seven of satiation! However, perhaps three branches are three years! It seems that he did not know through Nevua'h, for Brachos 55a learns from here that dreams are fulfilled like the interpretation! Perhaps he knew that in three days is Pharaoh's birthday, and every year he judges on his birthday.