The combinatorics of RNA plays a central role in biology. Mathematical
biologists have several commonly-used models for RNA: words in a fixed alphabet
(representing the primary sequence of nucleotides) and plane trees
(representing the secondary structure, or folding of the RNA sequence). This
paper considers an augmented version of the standard model of plane trees, one
that incorporates some observed constraints on how the folding can occur. In
particular we assume the alphabet consists of complementary pairs, for instance
the Watson-Crick pairs A-U and C-G of RNA.
Given a word in the alphabet, a valid plane tree is a tree for which, when
the word is folded around the tree, each edge matches two complementary
letters. Consider the graph whose vertices are valid plane trees for a fixed
word and whose edges are given by Condon, Heitsch, and Hoos's local moves. We
prove this graph is connected.
We give an explicit algorithm to construct a valid plane tree from a primary
sequence, assuming that at least one valid plane tree exists. The tree produced
by our algorithm has other useful characterizations, including a uniqueness
condition defined by local moves. We also study enumerative properties of valid
plane trees, analyzing how the number of valid plane trees depends on the
choice of sequence length and alphabet size. Finally we show that the
proportion of words with at least one valid plane tree goes to zero as the word
size increases. We also give some open questions.Comment: 15 pages, 10 figure
