Question $3$

$($a$)$ Using the properties of Red$-$Black Trees, show that the longest simple path from a node $x$ in a tree to a descendant leaf has a length at most twice that of the shortest simple path from node $x$ to a descendant leaf.

$($b$)$ Let $B$ be a Binary Search Tree, and $x$ be a number. Describe how to find the smallest number larger than $x$ in $B,$ even if $x$ is not necessarily found in $B.$

$($c$)$ Let $B$ be a Binary Search Tree, and $x$ be a number. Describe how to find the closest number to $x$ in $B.$ The closest means $mi{n}_i|x-a_i|,$ where $a_i$ is in $B.$