what is in the ashes that are left after charcoal burns?

Charcoal also originally comes from trees. The essential metal ions for plants are Mg^2+ (chlorophyll), Ca^2+ (cell walls) and Na^+ and K^+ for signalling. When the C is burned these metal ions remain I'm guessing as MgO, CaCO3, NaCO3, KCO3, phosphates, chlorides as the ash.

Potash: "The name derives from "pot ash", which refers to plant ashes soaked in water in a pot, the primary means of manufacturing the product before the industrial era." Potash is a K containing mineral.

http://en.wikipedia.org/wiki/Potash

Also look up soap manufacture in ancient times.

https://en.wikipedia.org/wiki/Soap#History_of_clea...

Is it basically the same as wood ashes? Yes, it is the mineral residue left from the wood.