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

Non-normal limiting distribution for optimal alignment scores of strings in binary alphabets