After a fire, the ashes have less mass and take up less space than the trees and vegataion before the fire. how can this be explained in terms of the law of conservation of mass?

Fire expends ENERGY which is a by-product of conversion of matter. 

The calories or therms of a fire equal a certain amount of mass. 

When MATTER (Electron) and ANTI-MATTER (Positron) collide they don't inhillate each other with nothingness, they always realse TWO gamma particles. 

The ENERGY of the TWO gamma particles CONSERVES the MASS of the ELECTRON and The POSITRON 

This is what Einstein meant by E=mc2 

ENERGY has physical properties than can be measured. 

These properties equal the mass that no longer exists after you measure the mass BEFORE and AFTER 

The AFTER MASS + the ENERGY MUST equal the BEFORE MASS to at least 99.999% 

You must take heat, light and other PARTICLES into consideration. 

Their MASS plus their ENERGY