We show that for every ϵ>0 there exists a sufficiently large d_0∈ℕ such that for every d≥ d_0, whp the random d-regular graph G(n,d) contains a T-factor for every tree T on at most (1-ϵ)d/log d vertices. This is best possible since, for large enough integer d, whp G(n,d) does not contain a (1+ϵ)d/log d-star factor.
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