Here is my quick answer:
1) If the student practices correctly throughout the week on a daily basis and does not master the material within a few days of the lesson, one lesson is enough per week.
However,
1) If your student is very young and tends to forget everything the next day (especially if there is no parent to work with them on a daily basis and practice together),
2) If you have difficulty enforcing regular practice,
3) If the student practices so much that they finish their assignment and get bored after 2 days,
4) If the lessons are not long enough to cover theory, duets, etc. because the student's attention span doesn't last past 30 minutes
5) If the student does not pay attention to correct technique and practices incorrectly all week
6) If you're in a rush to learn quickly
then a second lesson would greatly benefit the student.
From my observation in my daily teaching, these are the general reasons for my recommendations:
1) In the ideal situation, enough material can be covered during the lesson to challenge the student for the whole week with daily practice of 30 minutes to 1 hr. But given that piano lessons tend to be shorter these days (30 minutes), we often don't cover enough for the students that really practice that much. Also, in the ideal situation the student heeds the teachers pointers and advice on how to practice correctly with correct technique. I would rather a student not practice at all rather than practice the wrong way!
2) Some students just have a hard time absorbing everything in one 30 minute lesson. The younger students especially seem to wipe their memory clean the moment they get out of a lesson and on to the next activity. It helps if a parent is knowledgeable about the piano and can sit with the student every day and remind them of what was covered during the lesson and help them practice.
3) Many parents have a lot of difficulty enforcing daily practice of 20 minutes or more. If your student doesn't practice during the week, they completely make no progress by the next lesson and we end up using another lesson reviewing the same material. It almost makes more sense to just schedule an extra lesson or two during the week that you may consider as "supervised practice". Otherwise the parents rarely get to enjoy any return on the tuition they have paid for the weekly lessons.
4) With young students, 3 days practice can mean quite a lot of progress and having lessons twice a week really keep things moving along. I find it works really well with a highly motivated child that practices so much that they finish everything I assign within a couple of days and are at risk of getting bored and losing motivation.
5) Often, we limit lessons to 30 minutes because the student's productive attention span doesn't last that long. Unfortunately this does not give us enough time to go over technique, finger exercises, theory, duets, rhythm study, etc. I feel like many times my students are getting half the education I received because I can't possibly fit everything into 30 minutes. For this reason, I really do wish my 30 minutes students had lessons twice a week. Students with 1 hour lessons don't encounter this problem.
6) Many students don't really pay attention to HOW they should practice and are often a bit on the lazy side when it comes to correct technique which is the utter backbone of playing an instrument. I don't have this problem with adult students because the understand the importance of moving their fingers and wrist and all that the way I tell them too. Kids, on the other hand, don't really understand or care and often don't want to suffer through training their hands physically to play correctly. In this case, extra lessons can prevent bad habits or incorrect playing. Imagine if the student is playing something incorrectly. It would be better if the mistake was pointed out after half a week instead of after one full week.
7) Some students are in a rush to learn quickly, for whatever reason. Since I have observed that more lessons almost always equal faster progress, in this case I would also recommend extra lessons. However, if no one is in a rush and just enjoying the experience of learning piano, there is no need for extra lessons.
