Suffix trees are the most frequently used data structure in algorithms on words. Despite this, little is known about their behavior in a probabilistic framework. In this paper, we consider the depth of a compact suffix tree, also known as the PAT tree, under some simple probabilistic assumptions. In fact, for the case of an asymmetric alphabet, we prove that the limiting distribution for the depth in a PAT tree is the same as the limiting distribution for the depth in a PATRICIA trie, even though the PATRICIA trie is constructed over statistically independent strings. In other words, the limiting distribution for the depth in a PAT tree storing n suffixes is normal.