Full Souslin trees at small cardinals

A (Formula presented.) -tree is full if each of its limit levels omits no more than one potential branch. Kunen asked whether a full (Formula presented.) -Souslin tree may consistently exist. Shelah gave an affirmative answer of height a strong limit Mahlo cardinal (Formula presented.). Here, it is shown that these trees may consistently exist at small cardinals. Indeed, there can be (Formula presented.) many full (Formula presented.) -trees such that the product of any countably many of them is an (Formula presented.) -Souslin tree.