In all likelihood most siege engines would have been a melange of cut and scavenged woods, some chronicles testify to ships hulls and masts, and houses, torn apart. However, oak and beech are the most common references in chronicles from Charlemagne (8th c CE) to Froissart (14th c CE), but that would be in areas where it was plentiful from forests in France, England, Germany. Fir was a good replacement: a strong tree with height and stoutness. Ash as well would have been a good substitute for some parts under some stress where flex was acceptable, and it was common for wheels, although did not grow as big. Hornbeam for axels where available, pretty much the hardest wood in Europe although did not grow in size like other trees.

Once you get to the Levant there are stories where crusaders needed to travel miles around to find suitable woods, although the woods are not named. The plentiful pines would have been liable to snapping, although Arabic siege engines were made of cedar according to one chronicle of the 8th c CE.