Brainteaser 1742: Not-so-safe deposit

From The Sunday Times, 4th February 1996 [link]

The safe deposit boxes at our bank are arranged in a huge pyramid, the top of which is illustrated:

When the were first opened the top box had a quantity of gold coins placed in it and the rest were empty. One night a burglar broke into that box, emptied it, and (cleverly, he thought) instead of running off with the coins, he found the two empty boxes in the row below and put some of his haul in one and the rest in the other.

Word spread through the underworld about the bank’s lax security, and a sequence of burglaries took place. Each burglar broke into one box containing some coins, emptied it, found two empty boxes in the row below, placed some of his haul in one of these boxes and the rest in the other.

Ages later the manager of the bank opened up the top box and was horrified to find it empty. So he tried the boxes in the next row and found them empty too. He carried on like this until he found some coins in one particular row. In fact, no matter how many burglaries had taken place, there could not have been more empty rows at the top of the pyramid. And indeed if any fewer burglaries had taken place all those rows could not have been empty.

(a) How many empty rows at the top were there?
(b) How many burglaries were there?

This puzzle was included in the book Brainteasers (2002, edited by Victor Bryant). The puzzle text above is taken from the book.