The first ones look possibly like oak seedlings. Can you remove the lattice and try to pull them out?

The seedlings are oaks and mulberry.

If glyphosate will not kill them there probably is nothing else that will. If you could get to them enough to cut the stems below the leaves they will eventually die since they do need the nutrients those leaves manufacture to grow. Keeping them cut also means you do not need to spend money on poisons.

Turning a dark pot over the stumps will help to reduce the ability to regrow and more quickly exhaust any reserves of energy.

Coating the fresh cuts with salt &/or vinegar would cause damage as well, if not fatal. Skip the salt if the roots of other, desirable plants are nearby or "downstream" when it rains because it will persist in the soil for a while.

Pouring boiling water on stump/crown/immediate root zone would also be devastating. Might be an option on the mulberry but the oak looks like it sprouted back under a spot with little room to maneuver.

Glyphosate was never intended to kill woody plants - it was developed to control herbaceous weeds. For a woody based plants - like the 'weeds' above - you need something stronger like a brush killer with triclopyr as the active ingredient.

btw, homemade approaches like boiling water or vinegar will not work on these sorts of weeds. At best, all they do it damage/kill off the foliage but the root will remain undamaged and the plant will simply grow back. You either need to dig them out manually or apply a proper herbicide.

Yeah I was afraid this wasn't going to be so easy.

I've been cutting them back on a regular basis for a couple of years. They just seem to be getting bigger. I'll try the triclopyr this weekend. What's the best way to apply this herbicide? Should I cut the stems and rub it onto the fresh cut?

Basically, yes.

Here in the northwest, one brand of Blackberry & Brush Killer (triclopyr is the active ingredient) is in an applicator container with a small spout you flip open. Drip the product on the cut surface. The method will be described on the label.

It's very likely they will re-sprout (especially the mulberry), so plan to repeat the treatment as needed. Perhaps several times.

I see a lot of people cutting back tree weeds like these and then letting too much time pass before they cut them back again.During that in between time the plants grow back a lot of leaves which send more energy to the roots and their problem only becomes worse.You really have to keep on top of it and try not to allow any new top growth at all.If the plant can't photosynthisize then the roots will starve and weaken,and eventually die.Look out for these tenacious tree seedlings like mulberry,elm, buckthorn,and oak and yank them out as soon as you spot them because if not caught early they are impossible to pull out.