Why does a lightsource postpone the time it takes for the leaves falling from a birch tree?
It's to do with the chemistry taking place in the system, and then in turn the balance of energy resources.

Leaves contain chlorophyll, a molecule that traps light energy. It is large molecule containing a long carbon chain and a magnesium atom surrounded by 4 nitrogen atoms. In terms of a plants energy resources it is an 'expensive' molecule, but it is worth it as it is important to the plant.

The conversion of light energy by plants, using chlorophyll, is called photosynthesis. The chemistry of photosynthesis using chlorophyll is complex, but essentially Carbon Dioxide (CO2) from the air is combined with Water (H2O) from the roots to produce sugar and oxygen. Excess sugar is transported to the roots and stored as starch, and some sugar is used to keep the living cells throughout the plant alive. They need energy and this gained through respiration. Again, the actual chemistry is quite complex, but simply put the cells take in Oxygen from the air and 'burn' the sugar to release energy, this gives off Carbon Dioxide and Water. (There is plenty of resources and explanation on the actual mechanism - search for "ATP and ADP in plants")

Notice that it starts with carbon dioxide and water and ends with carbon dioxide and water, the energy is held within the chemical bonds.

When there is plenty of light (and water) the rate of photosynthesis, and therefore the rate of energy production, is high. The plant is producing excess energy, and the leaves are 'worth' the energy needed to keep them alive and functioning. Note that temperature also has an influence. The warmer the temperature (to a point) the greater the rate of photosynthesis.

So, in temperate climates with markedly different day lengths and temperatures between the summer and winter seasons some plants have evolved to be deciduous and shed their leaves in the autumn. The point at which this will start to happen will be when the day length shortens, and the temperature drops, and so when the amount of energy that the leaves are able to produce starts to balance the amount of energy being used to maintain the leaf.

If a plant has an artificial light source then the leaves will continue to be able to produce more energy than they use and so will be retained for longer.