Below is a step$-$by$-$step analysis of the Union operation scenario presented in the problem by applying the "Nodes Count Based Union Method" and "path compression". The goal in question is to find the "number of manipulations of pointer" and the "height of the tree that is finally formed" that occur when performing the Union operation in the given order with each of the first eight elements a$,$ b$,$ c$,$ d$,$ e$,$ f$,$ g$,$ and h being a singleton set. Pointer manipulation is the act of setting or changing the parent pointer once.

Initial state

Sets: $\{$a$\},\{$b$\},\{$c$\},\{$d$\},\{$e$\},\{$f$\},\{$g$\},\{$h$\}$

Size of each set $=1$

Each element forms a tree $($height $0)$ with itself as its root

Operational order:

union$($a$,$b$)$

union$($c$,$d$)$

union$($e$,$f$)$

union$($g$,$h$)$

union$($b$,$c$)$

union$($h$,$e$)$

union$($a$,$e$)$

Rules for each union$($x$,$y$)$:

Find root$($x$)$ and root$($y$)$ and merge the two trees.

A root in a tree with a smaller number of nodes $($size$)$ goes under a root in a larger tree.

If the sizes are the same, make root$($x$)$ a child of root$($y$).($i$.$e$.,$ if it is a uniformity, the root of x in union$($x$,$y$)$ always goes below the root of y$.)$

Apply path compression in the find process.