Approximating infinite graphs by normal trees

We show that every connected graph can be approximated by a normal tree, up to some arbitrarily small error phrased in terms of neighbourhoods around its ends. The existence such approximate trees has consequences both combinatorial and topological nature. On the side, we spanning tree as soon it locally at each end; i.e., only obstruction for having is an end which none tree. space ?(G), well spaces |G|=G??(G) naturally associated with G, are always paracompact. This gives unified short proofs number results Diestel, Sprüssel Polat, answers open question about metrizability Polat.