Packing large trees of consecutive orders

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 $k\le 5$.