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Can't help with the pricing - but I'd guess cedar will outlast oak given our weather.
Jim
Demonstrating every day that enthusiasm cannot overcome a lack of talent!
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You're right that white oak will be stronger, but I think cedar would work fine, too, as long as the cross sections aren't really dinky. I've used redwood in 3/4" cross section with no problems, and it has similar strength to cedar. The grain of cedar won't be as pronounced as it weathers if that's a consideration. Black locust and red mulberry would be even better choices, and Ipe' or Cumaru even better than that.
John
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Both very survivable for a long time in that use, IF it is off the ground at the bottom.
Oak much stronger, and remember if you are talking
Trellis it usually contains some amount of the very thin basket weave. If not then either is as strong as you will need. If it has trellis the WO will be much stronger.
Cedar, maybe cheaper, really depends where you live, and are you weaving from stock, or buying ready made? WO in the rough can be almighty cheap.
Worst thing they can do is cook ya and eat ya
GW
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Why do you want to half lap it? A whole lot of work and, if placed on a wall or fence non-lapped is functionally better because it is easier to tie in plants and gives them more space to twine through. For me, the 3D texture of non-lapped trellis is also more visually pleasing. It may well also be longer lasting because there is less end grain for water to be trapped in.
Jim