  Kashiwara's crystal graphs have a natural monoid structure that arises by
identifying words labelling vertices that appear in the same position of
isomorphic components. The celebrated plactic monoid (the monoid of Young
tableaux), arises in this way from the crystal graph for the $q$-analogue of
the general linear Lie algebra $\mathfrak{gl}_{n}$, and the so-called Kashiwara
operators interact beautifully with the combinatorics of Young tableaux and
with the Robinson--Schensted--Knuth correspondence. The authors previously
constructed an analogous `quasi-crystal' structure for the related hypoplactic
monoid (the monoid of quasi-ribbon tableaux), which has similarly neat
combinatorial properties. This paper constructs an analogous `crystal-type'
structure for the sylvester and Baxter monoids (the monoids of binary search
trees and pairs of twin binary search trees, respectively). Both monoids are
shown to arise from this structure just as the plactic monoid does from the
usual crystal graph. The interaction of the structure with the sylvester and
Baxter versions of the Robinson-Schensted-Knuth correspondence is studied. The
structure is then applied to prove results on the number of factorizations of
elements of these monoids, and to prove that both monoids satisfy non-trivial
identities.
