We prove that for any pair of constants $��>0$ and $��$ and for $n$ sufficiently large, every family of trees of orders at most $n$, maximum degrees at most $��$, and with at most $\binom{n}{2}$ edges in total packs into $K_{(1+��)n}$. 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.

38 pages, 2 figures; suggestions by an anonymous referee incorporated; accepted to Israel J Math