An approximate version of the tree packing conjecture

We prove that for any pair of constants ε > 0 and ∆ and for n sufficiently large, every family of trees of orders at most n, maximum degrees at most ∆, and with at most (n2) edges in total packs into K(1+ε)n. This implies asymptotic versions of the Tree Packing Conjecture of Gy´arf´as from 1976 and a tree packing conjecture of Ringel from 1963 for trees with bounded maximum degree. A novel random tree embedding process combined with the nibble method forms the core of the proof.