Reply by tenontim

The reason they used a lap joint instead of mortise and tenon, is because of the way the chains that it hangs from are attached. If you use a m&t joint, the through tenon would be considerably thinner than the way that it’s designed.
I will also second the remark by crank49, about the cedar not having the strength of the white oak. You might want to consider making the bottom frame out of oak and the rest out of your cedar. Then stain it to get the colors close.

-- Tim-- http://www.tmuli.com