  We classify extremal traces on the seven direct limit algebras of noncrossing
partitions arising from the classification of free partition quantum groups of
Banica-Speicher (arXiv:0808.2628) and Weber (arXiv:1201.4723). For the
infinite-dimensional Temperley-Lieb-algebra (corresponding to the quantum group
$O^+_N$) and the Motzkin algebra ($B^+_N$), the classification of extremal
traces implies a classification result for well-known types of central random
lattice paths. For the $2$-Fuss-Catalan algebra ($H_N^+$) we solve the
classification problem by computing the \emph{minimal or exit boundary} (also
known as the \emph{absolute}) for central random walks on the Fibonacci tree,
thereby solving a probabilistic problem of independent interest, and to our
knowledge the first such result for a nonhomogeneous tree. In the course of
this article, we also discuss the branching graphs for all seven examples of
free partition quantum groups, compute those that were not already known, and
provide new formulas for the dimensions of their irreducible representations.
