We prove that for any pair of constants epsilon > 0 and Delta and for n sufficiently large, every family of trees of orders at most n, maximum degrees at most Delta, and with at most ((n)(2)) edges in total packs into K(1+ epsilon)n. This implies asymptotic versions of the Tree Packing Conjecture of Gyarfas 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.
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