Packing the largest trees in the tree packing conjecture

The famous tree packing conjecture of Gyárfás from 1976 says that any sequence of trees T_{1},\ldots,T_{n} such that |T_{i}|=i for each i\in [n] packs into the complete n -vertex graph K_{n} . Packing even just the largest trees in such a sequence has proven difficult, with Bollobás drawing attention to this in 1995 by conjecturing that, for each k , if n is sufficiently large then the largest k trees in any such sequence can be packed into K_{n} . This has only been shown for k\leq 5 , by Żak, despite many partial results and much related work on the full tree packing conjecture. We prove Bollobás’s conjecture, by showing that, moreover, a linear number of the largest trees can be packed in the tree packing conjecture.