Bollobás's consecutive-order tree packing conjecture

Let $k\geq 1$ and let $T_n,T_{n-1},\dots,T_{n-k+1}$ be trees, where $T_{n-j}$ has $n-j$ vertices. The trees pack into $K_n$ when they can be found as pairwise edge-disjoint subgraphs of the complete graph $K_n$. Bollobás's conjecture. For every $k\geq 1$ there is an $n_0(k)$ such that if $n>n_0(k)$, then any set of $k$ trees $T_n,T_{n-1},\dots,T_{n-k+1}$ such that $T_{n-j}$ has $n-j$ vertices pack into $K_n$. This weakens Gyárfás's Tree Packing Conjecture, which asks for packing trees of every order from $1$ through $n$; the paper confirms the conjecture for certain trees and, as a consequence, for $k\leq 5$.