Packing trees of bounded diameter into the complete graph

Let d be a positive integer. We prove that there exists a constant c = c(d) such that if T1, . . . , Tn is a sequence of trees such that |V (Ti)| = i, diam(Ti) ≤ d + 2, and there exists xi ∈ V (Ti) such that Ti−xi has at least (1− c)(i− 1) isolated vertices, then T1, . . . , Tn can be packed into Kn. This verifies a special case of the Tree Packing Conjecture. We then prove that if T is a tree of order n +1 and there exists x ∈ V (T ) such that T−x has at least n− √ n/8 isolated vertices, then 2n + 1 copies of T may be packed into K2n+1. Finally, we show that there exists a constant c′ = c′(d) such that if T is a tree of order n + 1, diam(T ) ≤ d + 2, and there exists x ∈ V (T ) such that T − x has at least (1− c′)n isolated vertices, then 2n+1 copies of T may be packed into K2n+1. The last two results verify special cases of Ringels conjecture.