# Exercise 13.1.2

Draw the red-black tree that results after TREE-INSERT is called on the tree in Figure 13.1 with key $36$. If the inserted node is colored red, is the resulting tree a red-black tree? What if it is colored black?

The new element is going to be the gray one:

If we color it red, it will violate property 4, that is, red nodes need to have black children. In this case, it's parent, 35 is red, so it must be black.

If we color it black, it will violate property 5. The path to the descendants of 36 from the root will have 4 black nodes, but the one to the descendants of 39 will have 3.

That is, TREE-INSERT does not produce a valid red-black tree in this case, regardless of how we color the node.