I am no expert on folklore, but I presume rather than having anything to do with the goddess Diana, the importance of the oak tree in folk music stems from the fact that in the British Isles and probably much of Western Europe, it tended to be the dominant tree in much of the countryside when it was better wooded than today, and as it can live a long time (several hundred years) and was very important for timber (for construction of buildings, shipbuilding, etc) it was inevitable that it would become important in folklore. However in certain areas, depending on soil, climate, altitude and history of land use, elm, lime, pine, hazel, ash, etc, would be the dominant tree.

In parts of Ireland hazel was more important, which possibly explains occasional reference to hazel in traditional songs, eg "Flower of the hazel glade" in "Eileen Aroon". However I can think of one song "The Old Oak Tree" apparently from the North of Ireland, based on a murder where the female victim was found buried beneath "an old oak tree". I have been trying to find the words of this song (which appeared on the first Boys of the Lough LP, but so far no-one has sent me the same version). Reference is made to a "Squire McCollum" as the murderer (the victim being only known as "Betsy") so perhaps it relates to a specific murder.

Incidently some people may associate druids with mistletoe and oak trees, but apparenty mistletoe is more associated with apple trees.