  A family of Lempel-Ziv factorizations is a well-studied string structure. The
LZ-End factorization is a member of the family that achieved faster extraction
of any substrings (Kreft & Navarro, TCS 2013). One of the interests for LZ-End
factorizations is the possible difference between the size of LZ-End and LZ77
factorizations. They also showed families of strings where the approximation
ratio of the number of LZ-End phrases to the number of LZ77 phrases
asymptotically approaches 2. However, the alphabet size of these strings is
unbounded. In this paper, we analyze the LZ-End factorization of the
period-doubling sequence. We also show that the approximation ratio for the
period-doubling sequence asymptotically approaches 2 for the binary alphabet.
