Exact Enumeration of Word Trees with a Singleton Letter

We give exact counts of word trees whenever the defining word contains a letter exactly once. A word tree is an assignment of independent uniform transition maps whose composite has exactly one cyclic point. Conditioning on the singleton-letter map reduces the count to a two-point collision probability for the remaining word. A finite recursion on equality patterns of two trajectories computes this probability exactly, with a state space depending on the word rather than n. For two maps, T_AB(n)=(2n-1)n^(2n-2) and T_AAB(n)=(3n^3-3n^2-n+2)n^(2n-4), for n>=2; all A^rB words, rotations and positive powers are covered. The five-page proof credits the classical weighted Cayley/Pitman mapping identity and Martinsson's conditioning construction. These formulas answer Chapuy's explicitly proposed AB and AAB cases. The general enumeration, total-variation questions and neighboring permutation questions bundled into UnsolvedMath OWR-14299904-011 are not solved; the whole dataset record remains partially solved. Source code and exact checks accompany the paper. This is an unrefereed preprint developed with AI assistance, without formal proof-assistant verification or an absolute-priority claim.