Does random tree puzzle produce Yule-Harding trees in the many-taxon limit?

It has been suggested that a random tree puzzle (RTP) process leads to a Yule-Harding (YH) distribution, when the number of taxa becomes large. In this study, we formalize this conjecture, and we prove that the two tree distributions converge for two particular properties, which suggests that the conjecture may be true. However, we present statistical evidence that, while the two distributions are close, the RTP appears to converge on a different distribution than does the YH. By way of contrast, in the concluding section we show that the maximum parsimony method applied to random two-state data leads a very different (PDA, or uniform) distribution on trees.