I was in the SCA back in the day!
The 12th c. is part of the High Middle Ages. New technologies and population expansion were the highlights of the age. Knights were armoring up in response to a greater use of the crossbow. Most of the population were rural peasants (90%) that labored under the feudal system.
On the continent, a bow from this time period would most likely be used for hunting. A simple D-bow made from Yew (in the south) or Elm (in the north) would be normal. Faceted cross sections, fancy horn nocks, and recurved tips would not be common. Nocks would probably be cut into one side of each tip. I've seen a few pictures of tips that were bent "backwards", or deflexed, but this feature seems to be mostly an upper limb "thing" that was built into the bow for an unknown reason. Anyway, there are no surviving examples of German bows from the 12th century, as far as I know, so we have to play it safe and go with the simplest design: Man-sized, bendy handle, D-bow with circular or oval cross section. 50lb pull. Side nocks. Linen or hemp string. Hazel coppice arrows with tanged or socketed leaf-shaped iron points and goose feather fletching. Spiral linen cord wrap over brightly colored shaftment. Quiver of cloth attached at the belt.
Remember to aim for the horse and not the knight!