Pairwise ordered tree alignment are combinatorial objects that appear in RNA secondary structure comparison. However, the usual representation of tree alignments as supertrees is ambiguous, i.e. two distinct supertrees may induce identical sets of matches between identical pairs of trees. This ambiguity is uninformative, and detrimental to any probabilistic analysis.#R##N##R##N#In this work, we consider tree alignments up to equivalence. Our first result is a precise asymptotic  enumeration of tree alignments, obtained from a context-free grammar by mean of basic analytic combinatorics. Our second result focuses on alignments between two given ordered trees $S$ and $T$. By refining our grammar to align specific trees, we obtain a decomposition scheme for the  space of alignments,  and use it to design an efficient dynamic programming algorithm for sampling alignments under the  Gibbs-Boltzmann probability distribution. This generalizes existing tree alignment algorithms, and opens the door for a probabilistic analysis of the space of suboptimal RNA secondary structures alignments.
['Cedric Chauve', 'Julien Courtiel', 'Yann Ponty']
Counting, generating and sampling tree alignments