If you don't mind me saying, you are making two mistakes here.

First, you are assuming that roads did not exist in medieval Europe. This is categorically false. The remnants of various Roman paved roads survived in Britain, France, Spain, Italy, etc and continued to serve as thoroughfares. In addition to these, however, the countryside would have been laced with deer trails, foot paths, wagon roads, etc. These occur more or less spontaneously as a result of people and animals living in an area; even if the government wasn't cutting and paving proper roads, it was not a trackless wilderness.

Second, you are assuming that the typical medieval forest would have been extremely dense. This is understandable, as most modern forests are like this. But medieval forests were largely old growth, so this is not very likely to have been the case. In very old forests, the trees tend to be large but widely spaced and with very little underbrush between them, leaving considerable open ground for maneuver.

This is more of an addendum, but unless one were actively engaged in combat, there would be little to no reason to ride at speed through dense forest. Riding horses hell for leather will result in blown, useless horses, which is why cavalry rarely broke into a full charge until they were quite close to the enemy.

Archaeologist of combat reporting in:)

In general, cavalary battles were very, very rarely fought in the woods. Not just because the risk of crashing (rather the rider getting slammed into low hanging branches than the horse ramming a tree), but also because the woods provide great opportunity for the infantry and crossbowmen to ambush riders.

Atop of that, remember that the woods of pre-industrial times were far less "orderly" than modern ones. The ground between trees would be uneven, swampy, and full of holes big enough for a horse to break a leg. The trees would not be planted in neat rows, but grow randomly and at haphazard angles. Simply put, Medieval European woods were just as dense, and wild as Amazonian jungle.

The wars between the Polish-Lithuanian state and the Teutonic Order (especially those between 1409 and 1430ties) provide a lot of examples of cavalary getting ambushed and sandwitched between dense woods and a mass of infantry, and then decimated.

One example would be the Battle of Chojnice, when Polish cavalary was routed and funneled through a thin strip of free ground between masses of trees, and ran straight into a swamp.

The reversed situation happened during the Battle of Swiecino, when it was the Teutonic cavalary that got ambushed by Polish crossbowmen and infantry, and massacred (being tightly packed on a thin road between walls of thick woods made them a perfect target). Similarily to the example above, remaining ritter of the Order prefered to try their luck through the swamp than through the woods (most of them either got stuck, drowned or got chased down and killed).

So, to recap: While horses were not likely to simply ram into a tree, cavalary was still very ineffective in the woods, and avoided it out of the (reasonable) fear of an ambush or even accidental injury, not to mention the inability to gain speed and momentum.

Sources:

M. Biskup, The Thirteen Years War,

Robards, Brooks. The Medieval Knight at War