Term Test One Evaluation

After the Friday's lecture, the class had their term test one papers back. The term test consists of three questions.
The first question was about using a mathematical induction to prove a statement. The question was very similar to the tutorial question so the proving was extremely simple to write out. However, I lost a mark for not stating P(n). I thought that P(n) is basically the research question so I did not state the P(n) for this question. Unfortunately, the thought was wrong. I was right that P(n) is not equal to the research question but  P(n) comes from the research question. It was a basic mistake but taking off a mark out of 5 is too much. I admit that I deserve a mark deduction but losing 20% of a question is too much for a question when the question is about proving. From now on, I write out my P(n) before proving anything even though it is so obvious from the question.
The second question was about proving that a non-empty binary tree at height n has at most 2 to the power of n - 1. Like the first question, it was dealt during a tutorial so writing out a proof was an easy part. However, I lost another mark from stating P(n). For this time, I stated P(n) but I restricted the domain of the P(n). At first, I couldn't accept this deduction because the domain I stated is correct anyways. I asked my professor and the answer I got from him was kind of ambiguous to me. He first mentioned that I should never restrict the domain of P(n) because the domain is restricted later stating the P(n). However, logically, restricting twice, inside of P(n) and outside of P(n), does not change the statement. It would be inefficient but the statement would still be correct anyways. Second, he told me that a statement 'for a positive integer n' is equal to 'for all positive integer n' in this course so stating more than one 'n's is wrong. I still  could not understand this because 'a' and 'all' are different in the first place and we are not learning a so-called 'new English' for proofs. I think this is one thing that I need to accept anyways even though it does not really come to my mind because it is like a rule in order to show a proof. I will make this sure in my mind and never make this mistake again.
The third question was about the structural proof. I knew that it was in the midterm topics but did not expect this to come out because we did not have any tutorials dealt with this proof. Unfortunately, we had this proof and I got a bad mark on this one. For the next midterm and the final, I will cover everything that is in the topic list.