Q:

Why are there fewer stages in the turbine than in the compressor of a turbine engine?

I'm a student interested in aviation. I'd like to understand why the turbine section has fewer stages than the compressor section in a jet engine.
What would be the effects of adding stages to the turbine, or to the compressor? How is the optimal number of stages determined?

A:

Very simply, a turbine has fewer stages than the compressor, because the flow in the turbine is going in the direction of reducing pressure - the air wants to go that way naturally. But a compressor has to make the air flow in the direction of increasing pressure. If this is done too aggressively (attempting too much pressure rise per stage), the air flow on the blade seperates, and stalls. In severe situations, the flow can completely reverse direction and exit the intake.

A:

It's well-known that the designers always desire to less weight and less complexity, so if there isn't the need to add the stages to the turbine they will not add auxilary stages. Such, the problem is why the compressor need much more stages to achieve some compression ratio  than the number of stages of the turbine,simply to say, with somewhat less (numerically) expanding ratio. If exactly, the expanding process is continued in the exhaust nozzle. But this isn't the main cause of the less number of stages of a turbine. 
The explanation is following.
Every stage has a definite max loading coefficient that determinaned by aerodynamical possibility (certainly, w/o significant losses) to change flow direction on a stage (of a turbine or a compressor). The aerodynamical possibility of a hot gas flow with very high temperature (i.e. with very high sound speed and very less density, conseequently,  the flow is subsonic and with high Re number ) is much more than the possibility of the cold gas flow with  higher density  on a compressor, because the cold flow is with lower sound speed, and usually the flow is supersonic or high subsonic, and futhermore with much lower Re number (more influence of viscousity). It causes great difficulties for significant change of flow direction w/o essential losses.
And it's necessary to add a very important issue that the gas flow in the compressor moves towards the zones with higher pressure (pressure gradient is positive) that may lead to flow separation (contrary flow in boundary layer), while in the turbine the negative pressure gradient exists that is prevent factor for flow separation.
That is why the turbine blades are made thicker with great curvature and more "streamline" than the compressor blades. 
Therefore, a turbine 
stage has much more loading coefficient than a compressor stage and conseequently a turbine may be made with less number of stages.      

