On the example of the Fibonacci chain we show how to efficiently generate infinite words defined by substitutions. We introduce substitution trees and we present an algorithm that uses them to generate such infinite words. We use a stack to generate the words in a completely different manner than in traditional breadth-first tree traversals. We show that the algorithm has O(log n) space complexity and O(n) time complexity, where n is the length of the generated word. The aperiodic Fibonacci chain is used for construction of aperiodic pseudorandom number generators.