A finite subset of the natural numbers is weak-Schreier if min S ≥ |S|, strong-Schreier if min S > |S|, and maximal if min S = |S|. Let M_n be the number of weak-Schreier sets with largest element n and let (F_n)_n≥-1 denote the Fibonacci sequence. A finite set is said to be Zeckendorf if it does not contain two consecutive natural numbers. Let E_n be the number of Zeckendorf subsets of {1,2, …, n}. It is well-known that E_n = F_n+2. In this paper, we first show four other ways to generate the Fibonacci sequence from counting Schreier sets. For example, let C_n be the number of weak-Schreier subsets of {1,2,…,n}. Then C_n = F_n+2. To understand why C_n = E_n, we provide a bijective mapping to prove the equality directly. Next, we prove linear recurrence relations among the number of Schreier-Zeckendorf sets. Lastly, we discover the Fibonacci sequence by counting the number of subsets of {1,2,…,n} such that two consecutive elements in increasing order always differ by an odd number.