Packing large trees of consecutive orders

Abstract: A conjecture by Bollob\'as from 1995 (which is a weakenning of the famous Tree Packing Conjecture by Gy\'arf\'as from 1976) states that any set of $k$ trees $T_n,T_{n-1},\dots,T_{n-k+1}$, such that $T_{n-i}$ has $n-i$ vertices, pack into $K_n$, provided $n$ is sufficiently large. We confirm Bollob\'as conjecture for trees $T_n,T_{n-1},\dots,T_{n-k+1}$, such that $T_{n-i}$ has $k-1-i$ leaves or a pending path of order $k-1-i$. As a consequence we obtain that the conjecture is true for $k\leq 5$.