A mulberry (supposedly) planted by the bard, cut down in 1756, and turned into many more keepsake items than one tree could provide.
Mulberry trees came to Stratford-upon-Avon in the early 17th century, when King James imported thousands of seedlings from central Asia to England in a bid to get a domestic silk industry going. Silk moths live on mulberry leaves– but prefer white mulberries, while the trees imported were the the pie-making black mulberry type. Regardless, the trees were in fashion and it was considered a bit of a statement to have one of these new fruits in your garden.
It isn't known if Shakespeare actually planted the tree himself, but scholarship accepts it as likely. He certainly mentions mulberries in more than one play. By the 1700s, the tree was a pilgrimage site– tourists would come into town asking to see it. The contemporary owner of the house, the Reverend Francis Gastrell, got so fed up with the visitors that he cut the tree down in 1756. (He was run out of town on a tax issue soon after.)
The wood of the mulberry was kept a while and eventually used to construct keepsakes–memento mori items that (in their fervor) border on the catholic relic. Fragments of wood were embedded in book covers– carved into busts– used for clock housings and snuff boxes– turned into caskets and pocket scales. So many things were made out of the tree, in fact, that it is nearly impossible that they are all from the tree.
This same process has been replicated in the tree's living cuttings, which have circled the world to pop up in Shakespeare gardens, heritage sites, and in front of theatres. These too are "scions of the original mulberry" but few– if any– boast the age or the provenance that would make such a claim believable. Instead, these relics live in a world of fantasy, the kind of belief one might summon when watching a play, like in Coriolanus, when Volumnia says:
Thy stout heart
Now humble as the ripest mulberry
That will not hold the handling.
A small blog on the subject from the Shakespeare birthplace trust.