Was Hungary really a steppe when the Magyars invaded, though? The natural lowland vegetation for central Europe, including Hungary, is swamps and dense forests. Hungary does not get less rain than the surrounding regions, it's not at all an arid land.
Most of the Great Hungarian Plain was routinely flooded by the Danube and the Tisza, before these rivers were reined in in the mid of the 19th century. Supposedly, it sometimes reached levels when the two rivers touched in the middle.
So most of the plain was, well, a floodplain, with adequate light, meadowy forest vegetation. The rest is/was shurbland (check out the Wikipedia article on the Hortobagy National Park, that should give you a good idea on how that place looked thousand years ago), mostly because the two big rivers meant that a lot of smaller rivers flowed into them, diverting the irritation from places like the Hortobagy.
Also, you are confusing two things. There was a huge deforestation in 16-17th centuries in regions west of the Tisza/Danube, and in the highlands. This did indeed lead to the expansion of the Steppe into areas west of the Danube, but it did not create the steppe regions east of the Tisza. In fact, the main settling sites of the still nomadic Magyars were east of the Danube, same as with the Avars and other steppe people passing through here.
Permanent seditary occupation of the areas west of the Danube, like Esztergom Visegrád, or Tihany comes when the tribes switch from mostly nomadic behaviour with permanent wintering places to having permanent settling places and only going nomad in the summer months to herd cattle and raid germans.