$($a$)$ Describe the three different nodes in a regression trees, and which nodes have an associated query $($splitting rule$).$

$($b$)$ When growing a regression tree using CART, two types of splits are considered. Describe these splits and provide an example for each.

$($c$)$ Here is a car purchase classification tree for one customer on a budget:

Please answer c$,$ d$,$ e please

i$.$ How many root nodes, internal nodes, and terminal nodes are there?

ii$.$ What are the decisions to be made?

iii. What sort of car does the buyer want?

$($d$)$ A small tree $($with few internal nodes$)$ regression trees tend to under$-$fit the training data. Briefly explain what is meant by over$-$fitting and under$-$fitting the training data using regression trees.

$($e$)$ The predictive performance of a single regression tree can be substantially improved by aggregating many decision trees.

i$.$ Briefly explain the method of bagging regression trees.

ii$.$ Explain the difference between bagging and random forest.

iii. Briefly explain boosted regression trees.