Imagine a stick that makes you uglier every time you are hit with it. This is the ugly stick! The ugly tree is an exaggerated example: a tree made of ugly sticks. This is a well-known joke expression.

It's a joke expression "He's so ugly he must have been beat with the ugly stick"

Or you can say: "She's so ugly she must of fallen out of the ugly tree and hit every branch on the way down!"