This question is about the number of ways of combining an ordered
list of terms by repeating a single binary operation.
For example with three terms $a$, $b$ and $c$ there are
just two ways $((a\oplus b)\oplus c)$ and $(a\oplus (b\oplus c))$.
Suppose the binary operation $\oplus $ is just ordinary subtraction
and $a=12, \; b=7, \; c=5$ then $((a\oplus b)\oplus c)= 5 - 5 =0$
and $(a\oplus (b\oplus c))= 12 - 2 = 10$.
We are not concerned in this question with doing the 'arithmetic'
or with whether the answers are the same or different. We just want
to find out how many ways there are of combining the terms, or if
you like of putting brackets into the expression. Note that we need
the brackets because the answers may be different as in the
These two tree diagrams show the two cases for combining 3
Show that for four terms, (three binary operations) there are five
cases and find the number of cases for five terms and six
See also the problems
One Basket and