I ended up to a website that had nice looking cartoonish tree:
I'm looking a way to create something similar looking with Photoshop CS6.
What do I need to do to achieve such a curvy texture?
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In Photoshop CS6, you could try with the following (pardon my final result, it's done in a rush, but I hope you get the idea!):
Start by drawing a rectangular square shape with the
Rectangular Tool, and fill it with a dark color (in this case #573a28).
Pen Tool and in a new layer, draw some irregular shapes, the veins of the tree. In the third and fourth blocks you can see I actually only drew two of these shapes and repeated them with some rotation and size changes. You can duplicate them or redraw them, depending on your time and dedication. As mentioned in the other answers, you can actually use a real photo. Just put it behind your new layer with some opacity, and draw over them.
To make it look like your example, I applied some shadow to the veins using
Blending Options >
Drop Shadow. In my case it looks quite terrible, and I probably wouldn't use it. I'd use some highlights and shadows instead, redrawing some details in the sides of the veins (or you can repeat veins, but just in a lighter color).