how do I parenthesize the product of abcdef?? This is about catalan number and parenthesizing.
a)Determine the list of five 1's and five 0's that corresponds to each of these:


*

*(((ab)c)(d(ef))) = (what I did: 1110010110)

*(a(b(c(d(ef)))))) = (what I did: 1010101010)
3.((((ab)(cd))e)f) = (what I did: 1111001100)
b) find the way to parenthesize abcdef that corresponds to each given list of five 1's and five 0's


*

*1110010100 = (what I did: (((ab(c(de)f)

*1100110010 = (what I did: ((ab((cd)(e)f)

*1011100100 = (what I did: (a(((bc)(ef))


Did I do it correct? I seem to understand a bit but then I'm confused again. Anyone can please clarify this ??
 A: Question a)


*

*A letter is replaced by the empty word $\epsilon$

*An expression (XY) is replace by the word $\color{red}{1}u_x\color{blue}{0}u_y$ where $u_x$ is the word for $X$ and $u_y$ is the word for $Y$.


I add colors to understand where is the $1$ and the $0$ added at this step.
Example : (((ab)c)(d(ef)))


*

*(ab) is $\color{red}{1}\color{blue}{0}$

*((ab)c) is $\color{red}{1}10\color{blue}{0}$

*(ef) is $\color{red}{1}\color{blue}{0}$

*(d(ef)) is $\color{red}{1}\color{blue}{0}10$

*(((ab)c)(d(ef))) is $\color{red}{1}1100\color{blue}{0}1010$


Question b)
This is the inverse computation. 


*

*An empty word is replaced by a letter

*In a word $w$ find the first (leftmost) $0$ such that $w=\color{red}{1}u_x\color{blue}{0}u_y$ and $u_x$ contains the same number of $0$'s and $1$'s. Then replace by $(XY)$ where $u_x$ is the word for $X$ and $u_y$ is the word for $Y$


Example : $1110010100$ ($x$ is any letter)


*

*$\color{red}{1}11001010\color{blue}{0}$ will be cut into $11001010$ and $\epsilon$

*$\color{red}{1}10\color{blue}{0}1010$ will be cut into $10$ and $1010$

*$\color{red}{1}\color{blue}{0}$ will be cut into $\epsilon$ and $\epsilon$

*$\color{red}{1}\color{blue}{0}10$ will be cut into $\epsilon$ and $10$


So now


*

*$10$ is $(xx)$

*$1010$ is $(x(xx))$

*$11001010$ is $((xx)(x(xx))$

*$1110010100$ is $(((xx)(x(xx))x)$


Then replace $x$ by appropriate letters and $1110010100$ is $(((ab)(c(de))f)$
