  We consider two independent binary i.i.d. random strings $X$ and $Y$ of equal
length $n$ and the optimal alignments according to a symmetric scoring
functions only. We decompose the space of scoring functions into five
components. Two of these components add a part to the optimal score which does
not depend on the alignment and which is asymptotically normal.
  We show that when we restrict the number of gaps sufficiently and add them
only into one sequence, then the alignment score can be decomposed into a part
which is normal and has order $O(\sqrt{n})$ and a part which is on a smaller
order and tends to a Tracy-Widom distribution. Adding gaps only into one
sequence is equivalent to aligning a string with its descendants in case of
mutations and deletes. For testing relatedness of strings, the normal part is
irrelevant, since it does not depend on the alignment hence it can be safely
removed from the test statistic.
