  In this article, we study the relationship between notions of depth for
sequences, namely, Bennett's notions of strong and weak depth, and deep
$\Pi^0_1$ classes, introduced by the authors and motivated by previous work of
Levin. For the first main result of the study, we show that every member of a
$\Pi^0_1$ class is order-deep, a property that implies strong depth. From this
result, we obtain new examples of strongly deep sequences based on properties
studied in computability theory and algorithmic randomness. We further show
that not every strongly deep sequence is a member of a deep $\Pi^0_1$ class.
For the second main result, we show that the collection of strongly deep
sequences is negligible, which is equivalent to the statement that the
probability of computing a strongly deep sequence with some random oracle is 0,
a property also shared by every deep $\Pi^0_1$ class. Finally, we show that
variants of strong depth, given in terms of a priori complexity and monotone
complexity, are equivalent to weak depth.
