Draw Your Work:

Complete the task by hand on paper, ensuring that your diagrams are clear and legible. Once done, take clear pictures of your work and upload them to Blackboard for submission.

Steps to Complete:

$($a$)$ Insertion:

Start with an empty Red$-$Black Tree and insert the following keys in the given order:

$50,13,76,56,67,85,59,60,15,20,45,83,72,34,23,49,54,87,72,64.$

Ensure you follow the Red$-$Black Tree properties $($e$.$g$.,$ maintaining balance, appropriate color adjustments, and rotations$)$ during each insertion. Each correctly inserted node is worth one point.

$($b$)$ Deletion:

Starting from the tree created in part $($a$),$ delete the following keys:

$83,85,76,$ and $67.$

Carefully demonstrate how the tree adjusts to maintain the Red$-$Black properties after each deletion.