Question
So I have two unidentified new plants growing pretty fast. They both look like they could be trees. They popped up around 4-6 months ago. They are growing where I don't normally cut the grass and I'm wondering what they actually are and if they are worth keeping around.

If I keep them, I would want to replant them in a suitable place. One is growing kind of under the kids trampoline in the backyard. The other is growing on the side of the house, right next to the brick wall. The leaves for both are different and look like no other tree/shrub/bush around me, or at least with-in a few houses of me.

(Click image for larger view)
Here's the first one. This grew last spring, but died after a month or two from heat. Around March or April I think, I noticed something growing again and left it alone. This is how big it is now.
<a href="http://i79.photobucket.com/albums/j123/Titanos/IMG_20100816_184555.jpg" target="_blank"><img src="http://i79.photobucket.com/albums/j123/Titanos/th_IMG_20100816_184555.jpg" border="0" alt="Photobucket" ></a>
Close up of the leaf
<a href="http://i79.photobucket.com/albums/j123/Titanos/IMG_20100816_184546.jpg" target="_blank"><img src="http://i79.photobucket.com/albums/j123/Titanos/th_IMG_20100816_184546.jpg" border="0" alt="Photobucket" ></a>
Here is the one in the backyard. This one sprung up about the same time I beleive. Close up of the trunk. I used to have alot of weeds in this spot. I used weed killer safe for grass last year and no more weeds. Now this started growing but it didn't look like weeds. So I left this alone as well.
<a href="http://i79.photobucket.com/albums/j123/Titanos/IMG_20100815_174611.jpg" target="_blank"><img src="http://i79.photobucket.com/albums/j123/Titanos/th_IMG_20100815_174611.jpg" border="0" alt="Photobucket" ></a>
And its leaf
<a href="http://i79.photobucket.com/albums/j123/Titanos/IMG_20100815_174549.jpg" target="_blank"><img src="http://i79.photobucket.com/albums/j123/Titanos/th_IMG_20100815_174549.jpg" border="0" alt="Photobucket" ></a>

No flowers or berries. Just leaves so far. I have no other weeds in my yard or in my neighbors yards.

Any ideas? Let me know if you need different pictures to help ID it. Thanks

Answer
Well, your first one looks a lot like a red mulberry. It's definitely a tree of some kind, with that thick woody stem.

The second is giving me more trouble. It might be a "wildflower" as it's not as obviously woody. I haven't gotten far with large, alternate palmate leaves. It reminds me of a castor bean, but definitely isn't as the stem would be attached to the leaf blade differently. The fact that you've seen nothing but leaves so far implies it's perennial at the very least, and likely a shrub.

A location would help narrow it down a lot. Judging from the palm in the background of the first shot, you're likely on the Gulf or West Coast, assuming you're in the US, if not, all bets are off. Even so, I'm better with temperate plants.

Sorry I can't be of more assistance than this.