Pieter proposes a tree structure where only the headers of the parent chains need to be stored, while Mike argues that in order to ensure validation, everything must be validated. Pieter counters by stating that with his system, one can be sure to catch all transactions corresponding to specific blocks or addresses. This is important because filtering based on criteria can result in missed transactions. Additionally, Pieter points out that his proposed system has advantages for mining decentralization. However, Mike maintains that sidechains are not a solution for scaling up.