Scotch Broom, and English Ivy under Live Oaks?

It may be pretty but Scoth Broom is coming up everywhere in my garden and along the creek at the side of the house. Some of the plants have reached 8ft with 3in diameter trunks! The seedlings pull up easily but these "trees" are a problem. Will they sprout if I saw them off at the ground?

The ivy was planted 40 years ago, by the man who built the house, and now covers the banks on two sides of the house and is hoisting itself up the Live Oaks too. I was up the banking cutting it off the trees but stood on a bees nest, got stung and haven't been brave enough to go up there again. Is there an easy way of getting rid of the ivy from the banking without harming the oaks?

My guess is the broom would probably re-sprout if you cut it down. If you can't dig it up, I would cut it off at the base, then take something like Brush B Gone and paint it on the trunk where you cut. Not sure if that's strong enough to kill it, but it's worth trying. For the ivy, hand pulling is probably the way that's least likely to damage the oak, again if you can break it off in places and paint the broken stems with Brush B Gone that'll probably speed up the process of killing it. You can have people come in and dig it out, but that'll probably damage the oak roots. I've found herbicides of any sort to be ineffective on ivy unless combined with pulling out as much of it as you can by hand, maybe someone else will have some better suggestions.