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Let $G=\mathop{A\ast B}\limits_C$ be an amalgamated product of finite rankfree groups $A$, $B$ and $C$. We introduce atomic measures and correspondingasymptotic densities on a set of normal forms of elements in $G$. We alsodefine two strata of normal forms: the first one consists of regular (orstable) normal forms, and second stratum is formed by singular (or unstable)normal forms. In a series of previous work about classical algorithmicproblems, it was shown that standard algorithms work fast on elements of thefirst stratum and nothing is known about their work on the second stratum. Inmain theorems A and B of this paper we give probabilistic and asymptoticestimates of these strata.

Amalgamated Products of Groups II: Measures of Random Normal Forms