How does the amount of eggs used in a cake affect how high it rises?

The eggs are to hold the cake together and it's the bicarbonate or baking powder that causes the cake to rise. Try 2 teaspoons of flax powder mixed with half a cup of water. This has great binding properties and is high in fiber and Alpha Lipoic Acid (ALA), an omega-3 fatty acid. A mashed banana is also a good substitute for eggs.