A remarkable result of Friedman and Pippenger  gives a sufficient condition on the expansion properties of a graph to contain all small trees with bounded maximum degree. Haxell  showed that under slightly stronger assumptions on the expansion rate, their technique allows one to find arbitrarily large trees with bounded maximum degree. Using a slightly weaker version of Haxell’s result we prove that a certain family of expanding graphs, which includes very sparse random graphs and regular graphs with large enough spectral gap, contains all almost spanning bounded degree trees. This improves two recent tree-embedding results of Alon, Krivelevich and Sudakov .
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