Sharp threshold for the appearance of certain spanning trees in random graphs

We prove that a given tree T on n vertices with bounded maximum degree is contained asymptotically almost surely in the binomial random graph \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}G(n,\frac{(1+\varepsilon) \log n}{n})\end{align*}\end{document} provided that T belongs to one of the following two classes: (1) T has linearly many leaves; (2) T has a path of linear length all of whose vertices have degree two in T . © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012