Solve this Questions taken from Algorithms, Fourth Edition chapter Balance search tree and page number $449.$

$3\mathrm{.}3\mathrm{.}1$ Draw the $2-3$ tree that results when you insert the keys E A S Y Q U T I O N in

that order into an initially empty tree.

$3\mathrm{.}3\mathrm{.}2$ Draw the $2-3$ tree that results when you insert the keys YLPMxHCRAES

in that order into an initially empty tree.

$3\mathrm{.}3\mathrm{.}3$ Find an insertion order for the keys S E A R C H X M that leads to a $2-3$ tree

of height $1.$

$3\mathrm{.}3\mathrm{.}4$ Prove that the height of a $2-3$ tree with $N$ keys is between ${|}_{{?}_{?}}lo{g}_3{N}_{{?}_{?}}|$

$\mathrm{.}63lgN($for a tree that is all $3-$nodes$)$ and $lg{N}_{{?}_{?}}|($for a tree that is all

$2-$nodes$).$

$3\mathrm{.}3\mathrm{.}5$ The figure at right shows all the structurally different $2-3$ trees with $N$

keys, for $N$ from $1$ up to $6($ignore the order of the subtrees$).$ Draw all the

structurally different trees for $N=7,8,9,$ and $10.$