If you want to multiply a pair of trees $(T,T')$ and $(G,G')$, put $T'$ on top of $G$, that is, identify their roots and then if there is a vertex with two carets 
$a\to (b,c), a\to (x,y)$, identify the edges $(a,b), (a,x)$ and $(a,c), (a,y)$. That is if two pairs of children  have the same parent, identify the pairs of children. After all these foldings are done, you get a finite binary tree $T''$, containing $T'$ and $G$ as rooted subtrees. Therefore $T''$ differs from $T'$ by a bunch of carets. Add the corresponding carets to $T$, get a tree $T'''$. The pair $(T''',T'')$ represents the same element as $(T, T')$. Similarly modify $G'$ and $G$ to obtain an equivalent pair of trees $(T'', G'')$. Then the product is $(T''', G'')$.  An easy procedure to get from a pair of trees (i.e. a "diagram") to a word in generators is desribed in chapter 5 of my book "Combinatorial algebra: syntax and semantics".