Phylogenetic trees are used to model evolution: leaves are labeled to represent contemporary species (``taxa""), and interior vertices represent extinct ancestors. Informally, convex characters are measurements on the contemporary species in which the subset of species (both contemporary and extinct) that share a given state form a connected subtree. Kelk and Stamoulis [Adv. Appl. Math., 84 (2017), pp. 34--46] showed how to efficiently count, list, and sample certain restricted subfamilies of convex characters, and algorithmic applications were given. We continue this work in a number of directions. First, we show how combining the enumeration of convex characters with existing parameterized algorithms can be used to speed up exponential-time algorithms for the maximum agreement forest problem in phylogenetics. Second, we revisit the quantity g2(T), defined as the number of convex characters on T in which each state appears on at least 2 taxa. We use this to give an algorithm with running time O(Φ n • poly(n)), where Φ ≈ 1.6181 is the golden ratio and n is the number of taxa in the input trees for computation of maximum parsimony distance on two state characters. By further restricting the characters counted by g2(T) we open an interesting bridge to the literature on enumeration of matchings. By crossing this bridge we improve the running time of the aforementioned parsimony distance algorithm to O(1.5895n • poly(n)) and obtain a number of new results in themselves relevant to enumeration of matchings on at most binary trees.