The Lempel-Ziv parsing scheme finds a wide range of applications, most notably in data compression and algorithms on words. It partitions a sequence of length n into variable phrases such that a new phrase is the shortest substring not seen in the past as a phase. The parameter of interest is the number M_n of phrases that one can construct from a sequence of length n. In this paper, for the memoryless source with unequal probabilities of symbols generation we derive the limiting distribution of M_n which turns out to be normal. This proves a long-standing open problem. In fact, to obtain this result we solved another open problem, namely, that of establishing the limiting distribution of the internal path length in a digital search tree. The latter is a consequence of an asymptotic solution of a multiplicative differential-functional equation often arising in the analysis of algorithms on words. Interestingly enough, our …