  When $k|n$, the tree $\mathrm{Comb}_{n,k}$ consists of a path containing
$n/k$ vertices, each of whose vertices has a disjoint path length $k-1$
beginning at it. We show that, for any $k=k(n)$ and $\epsilon>0$, the binomial
random graph $\mathcal{G}(n,(1+\epsilon)\log n/ n)$ almost surely contains
$\mathrm{Comb}_{n,k}$ as a subgraph. This improves a recent result of Kahn,
Lubetzky and Wormald. We prove a similar statement for a more general class of
trees containing both these combs and all bounded degree spanning trees which
have at least $\epsilon n/ \log^9n$ disjoint bare paths length $\lceil\log^9
n\rceil$.
  We also give an efficient method for finding large expander subgraphs in a
binomial random graph. This allows us to improve a result on almost spanning
trees by Balogh, Csaba, Pei and Samotij.
