Starting with the union$-$find data structure shown below, show the

sequence of union$-$find data structures that result from applying the

following operations $($in that order$)$:

union$(1,2)$union$(3,4)$union$(4,5)$union$(6,8)$union$(5,8)$

$$ union $(1,6)$ union $(7,9)$union$(10,11)$union$(11,9)$union$(1,11)$

Answer this question for each of the three following parts separately:

a$)$ The union$()$s are performed by height $($same as union$-$by$-$rank$)$ and find$()$s are simple;

b$)$ The union$()$s are performed by size and find$()s$ are simple;

c$)$ The union$()$s are performed by height and find$()$s use path com$-$

pression.

In your answer show the modified data structure after each union$()$

operation