$(1)$ Show that the longest simple path from a node $x$ in a red$-$black tree to a descendant leaf

has length at most twice that of the shortest simple path from node $x$ to a descendant leaf.

$(2)$ How the red$-$black trees after successively inserting the keys $50,42,30,17,25,6$ into and

initially empty red$-$black tree.

$(3)$ I would like to upgrade the RB tree shown below by inserting the number $180$ into it$.$

Show how this can be done such that after the insertion it remains to be a valid RB tree.

For full credit, you should show intermediate steps $($i$.$e$.,$ by showing how the tree is being

transformed into the final one$)$ by drawing a few trees.