Purely functional data structures are data structures that do not allow any modifications. Instead, all operations that would modify the data structure return a new data structure with the modification applied, leaving the original data structure unchanged. This immutability ensures thread safety and simplifies reasoning about code.

The benefits include:

• Thread safety: No need for locks in concurrent environments.
• Persistence: Old versions of data structures are preserved and can be reused.
• Simplicity: Easier to reason about and debug.
• Sharing: Memory can be shared efficiently between versions of data structures.

Persistence refers to the property of a data structure that preserves the previous version of itself when modified. This means that any update to the data structure results in a new version, while the old version remains unchanged and available for use.

Amortized analysis is a method for analyzing the time complexity of an algorithm over a sequence of operations, averaging the time taken over all operations. In the book, it is used to demonstrate that certain operations on data structures can be efficient on average, even if individual operations may be costly.

Lazy evaluation delays the computation of values until they are needed, which can improve performance by avoiding unnecessary calculations. It also allows for the creation of infinite data structures and can lead to more modular and compositional code.

Red-black trees are a type of self-balancing binary search tree that ensure the tree remains balanced after insertions and deletions. They are important in functional programming because they provide efficient, purely functional implementations of sets and maps with logarithmic time complexity for insertion, deletion, and lookup.

Binomial heaps are a type of heap (priority queue) that support efficient merging of two heaps. They are used in functional programming to implement priority queues with efficient operations for insertion, merging, and finding the minimum element.

The skew-binary number system is a numeral system that allows for efficient implementation of certain data structures, such as random-access lists. It provides a way to represent lists with efficient access and update operations, leveraging the properties of the skew-binary representation.

Real-time deques are double-ended queues that support constant-time operations for inserting and removing elements from both ends. They are designed to provide worst-case constant-time performance for all operations, making them suitable for real-time applications.

Catenable lists are lists that support efficient concatenation operations. Unlike regular lists, which require linear time to concatenate, catenable lists provide constant or logarithmic time concatenation, making them more efficient for certain applications.

Bootstrapped data structures are data structures that use instances of themselves to achieve better performance. They are useful because they can provide more efficient operations by leveraging the structure of the data itself, often leading to improved time complexity for certain operations.