Packing the largest trees in the tree packing conjecture

The famous tree packing conjecture of Gy\'arf\'as from 1976 says that any sequence of trees T₁,…,Tₙ such that |Tᵢ|=i for each i∈ [n] packs into the complete n-vertex graph Kₙ. Packing even just the largest trees in such a sequence has proven difficult, with Bollob\'as 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 Kₙ. This has only been shown for k≤ 5, by \.{Z}ak, despite many partial results and much related work on the full tree packing conjecture. We prove Bollob\'as's conjecture, by showing that, moreover, a linear number of the largest trees can be packed in the tree packing conjecture.