Not necessarily mistletoe (омела)! Another traditional English Christmas carol -- which is possibly older than the U.S. -- is The Holly and the Ivy, "Падуб и Плющ". An interesting thing about this carol is that the строфы (verses) are full of references to "Страсти Христовы" (blood, thorns, etc.), but the припев (refrain) is completely non-religious -- more like a New Year song than a Christmas song. It reminds me a little of "Прилетила, ластивочка...":
"Oh, the rising of the sun,
And the running of the deer!
The playing of the merry organ,
Sweet singing in the choir!"
In short, "among the leaves so green" can refer to many different varieties of "вечнозелёные растения".