Assume we are talking about such a tree for all subquestions in this part. Note, there are $15$ leaf nodes and $15$ possible values from $1-15$ so each value is used exactly once.

Hint: Tom has a few options in how to rearrange leaves to get max pruning. Try to derive constraints on the values that need to hold for max pruning to happen.

Use this tree with place holder letters to answer the questions below.

$($i$)[2$ pts$]$ If the value of the root is $8,$ which of the following leaf nodes are guaranteed to have value

$($iii$)[2$ pts$]$ If the value of the root is $8,$ which of the following leaf nodes are guaranteed to have value $8?$

$($iv$)[2$ pts$]$ If the value of the root is $6,$ which of the following leaf nodes are guaranteed to have value $6?$

$($v$)$ pts$]$ Which of the results $($root values$)$ are possible for Tom's rearranged game tree?

$$

$$

$$

$$

$5,64,59$

$8$

$13$

$12$

$11$

$$

$8$

$10$