Maybe Tennis shoe looks similar to running shoe, but there is a big different in the sole of the shoe. The sole of a tennis shoe is thicker and stronger. It's keep on your ankles when you brake on the court.
As I mentioned before there are four types of courts. Therefore there is a shoe for each type of court.
Here is an example for a teniis shoe:


There are many good comanies of sport shoe but not all of them sell tennis shoe. for instance: saucony.
It's very important to know the difference and to know which shoe are you going to buy.