An approximate version of the tree packing conjecture

We prove that for any pair of constants ɛ > 0 and Δ and for n sufficiently large, every family of trees of orders at most n , maximum degrees at most Δ, and with at most ( n 2 ) edges in total packs into . This implies asymptotic versions of the Tree Packing Conjecture of Gyárfás 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.