[The Squatter of Amon Rûdh]

Tolkien doesn't say that it did, and I think that he would have done had he imagined it. Perhaps it was left untended and untrimmed, but that would have done it no harm. Personally I think that Saruman would have been the last person to give the Old Forest a chance to intrude on the Shire, at least while he was living there. He had not enjoyed his last encounter with primeval woodlands at all. There's also the question of control: the High Hay was a barrier against entry into the Shire, and the more barriers there were around it, the easier it was for Saruman to keep track of comings and goings across its borders, and to curtail them if necessary.