  The famous tree packing conjecture of Gy\'arf\'as 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 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 $K_n$. This has only been shown for $k\leq 5$, 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.
