The number of subordinate cities depends on your rank. If you look in ranking it will tell you the requirements. The max is 9 including your main one

AS Stephen says you need to open them all. The higher the tavern the higher the chance of getting better generals. some say tavern 20 is sufficient - I usually wait for tavern 25 before I start general hunting. You can get lucky and a gold general may appear in your tavern as it auto refreshes every 4 hours.