Oh my.... this is one of my soap box subjects! you might regret this!! 
We moved this spring to a house with a good sized garden certainly big enough for 2 of us both with jobs and time share dogs we borrow every now and again, as we do with nephews and nieces! once we actully stood back and looked at what needed doing, we worked out that the people in before us did nothing appart from cut the grass for the 7 years they where in there.
We've been hacking and clearing all summer and finally we're down to just the ivy going along the back boundry "hedge" I use inverted commas because the hedge actually consists of 4 or 5 youngish damson trees that are covered in Ivy which then spreads all along the fence.
We decided to leave this bit until last because the birds where nesting in there. The branches of the Ivy are as thick as my arm which is ok but then there are all the off shoots even when you pull them out there are still bits under the ground and berries buried by the birds / squirrels it drives me nuts! not to mention the roots all through the back boarder! 
We've managed to get the 30 year old stuff down from the side of the house, the WHOLE side, I remember the houses being built and know when it went in, but now we have a 30 year old stump left... we've got stuff to drill holes and pour in!
So I suppose after my rant... (sorry about that!) the question is are you talking fairly young stuff which crops up occasionally or is it more of a tree with full on branches, because if you have as much as we've cleared, never mind have left to go I wouldn't fancy painting gel on all the leaves! 