  We prove that there is $c>0$ such that for all sufficiently large $n$, if
$T_1,\dots,T_n$ are any trees such that $T_i$ has $i$ vertices and maximum
degree at most $cn/\log n$, then $\{T_1,\dots,T_n\}$ packs into $K_n$. Our main
result actually allows to replace the host graph $K_n$ by an arbitrary
quasirandom graph, and to generalize from trees to graphs of bounded degeneracy
that are rich in bare paths, contain some odd degree vertices, and only satisfy
much less stringent restrictions on their number of vertices.
