you can work it out how deep the founds need to be, check out the table in "building near trees" in the NHBC guidelines (they are online somewhere)

you need to know the species of tree, work out distance (from tree to the found) over height (max height of the tree, it tells you in the table) and then look it up on the graph.

anything over 2.5m deep and Building Control want structural calcs for the footings, but dont worry they are only a few hundred quid (and the engineer might find you an easier more cost effective solution with some small piles or a reinforced raft foundation or something)

some people pull the trees out before they build, to get round all this, but in the case of fast-growing conifers this makes the soil conditions worse! and theres always nosey neighbours grassing to the council these days.

even doing it all yourself you will seriously struggle to do it for £5k.