AbstractA remarkable result of Friedman and Pippenger gives a sufficient conditionon the expansion properties of a graph to contain all small trees with boundedmaximum degree. Haxell showed that under slightly stronger assumptions onthe expansion rate, their technique allows one to find arbitrarily large trees withbounded maximum degree. Using a slightly weaker version of Haxell’s result weprove that a certain family of expanding graphs, which includes very sparse ran-dom graphs and regular graphs with large enough spectral gap, contains all almostspanning bounded degree trees. This improves two recent tree-embedding results ofAlon, Krivelevich and Sudakov ...