This is what the plant looks like

Looks like a walnut.

There's a Japanese walnut known as a heartnut. Both the shell and the edible nut inside are heart-shaped.

Here's Miller Nurseries, who'd like to sell you one:
http://www.millernurseries.com/cart.php?m=product_detail&p=1028

And general info from Wikipedia:
https://en.wikipedia.org/wiki/Juglans_ailantifolia

If you know anyone who does crafts, you might want to offer them the nuts. (I've no idea how troublesome they are to deal with, or how badly they stain.)

It's an odd thing to find growing on its own. I assume it was planted by a previous owner?

This post was edited by missingtheobvious on Wed, Sep 11, 13 at 17:42

A split Black Walnut reveals the seam and it is heart shaped. The shell is a large heart around 2 small lobes containing the nut meat. I think it is just a regular Juglans nigra.

Looks like black walnut to me. Might've been "planted" by a squirrel. They love those nuts.

The black walnuts I'm familiar with have oval shells, often a bit pointy on one end, but I wouldn't call them heart-shaped.

Depends on who's looking at them, I guess.