Pigeon Berry

Description:

Green leaves with a white flower and red berries. These plants were growing low to the ground under some oak and mesquite trees. They did not appear to be vines. However, in another area of the playground there is a vine that looks very similar also with red berries, but I have never seen it flower. It grows on the fence that surrounds our playground, and under an oak tree.

Notes:

I would really like some help with an i.d. After a little research I am thinking that perhaps the low growing plants might be pigeon berry, and the vine some type of snailseed vine.