To convert LAS into DEM / DSM I would recommend LAStools, specifically LAS2DEM, which is free to use for non-commercial use and not that expensive for professional use.
To convert DEM/DSM images into ASCII I use GDAL_Translate with AAIGRID driver or the QGIS raster save as
Convert your ground and model key points to MultiPoint using LAS to Multi Point (class codes 2 and 8 in the ASPRS spec) then do the same for your trees (low veg = 3, medium = 4 and high = 5), you might want to skip the low vegetation as it's probably noise.
To create a DEM from your multipoint make a terrain and import your ground/model key multipoints.
To make your tree DSM use Point to Raster with the shape.z as the source, this will not give complete coverage - but that's ok, there isn't trees everywhere.
Alternately you can create a LAS Dataset and filter the codes you are interested in, then use LAS Dataset to Raster to create your raster(s)... use TRIANGULATION for your DEM and BINNING (Cell Assignment Type = MAXIMUM) for your tree raster.
From here you can use IsNull on the tree raster to create a binary raster showing were trees are (and aren't) or Diff to calculate the tree height above the surface.