Data structures can also be recursive.

Trees are a recursive data structure because their definition is self-referential. A tree is a data structure which contains a piece of data **and references to other trees!**

Trees which **are referenced** by other trees are known as *children*. Trees which **hold references** to other trees are known as the *parents*.

A tree can be both parent and child. We’re going to write a recursive function that builds a special type of tree: a *binary search tree*.

Binary search trees:

- Reference two children
**at most**per tree node. - The “left” child of the tree must contain a value
**lesser than**its parent - The “right” child of the tree must contain a value
**greater than**its parent.

Trees are an *abstract data type*, meaning we can implement our version in a number of ways as long as we follow the rules above.

For the purposes of this exercise, we’ll use the humble Python dictionary:

bst_tree_node = {"data": 42} bst_tree_node["left_child"] = {"data": 36} bst_tree_node["right_child"] = {"data": 73} bst_tree_node["data"] > bst_tree_node["left_child"]["data"] # True bst_tree_node["data"] < bst_tree_node["right_child"]["data"] # True

We can also assume our function will receive a **sorted list of values** as input.

This is necessary to construct the binary search tree because we’ll be relying on the ordering of the list input.

Our high-level strategy before moving through the checkpoints.

**Base case**: The input list is empty

- Return
`"No Child"`

to represent the lack of node

**Recursive step**: The input list must be divided into two halves

- Find the middle index of the list
- Store the value located at the middle index
- Make a tree node with a
`"data"`

key set to the value - Assign tree node’s
`"left child"`

to a recursive call using the**left half of the list** - Assign tree node’s
`"right child"`

to a recursive call using the**right half of the list** - Return the tree node

### Instructions

**1.**

Define the `build_bst()`

function with `my_list`

as the sole parameter.

If `my_list`

has no elements, return “No Child” to represent the lack of a child tree node.

This is the base case of our function.

The recursive step will need to remove an element from the input to eventually reach an empty list.

**2.**

We’ll be building this tree by dividing the list in half and feeding those halves to the left and right sides of the tree.

This dividing step will eventually produce empty lists to satisfy the base case of the function.

Outside of the conditional you just wrote, declare `middle_idx`

and set it to the middle index of `my_list`

.

Then, declare `middle_value`

and set it to the value in `my_list`

located at `middle_idx`

.

Print “Middle index: “ + `middle_idx`

.

Then, print “Middle value: “ + `middle_value`

You can use `.format()`

or addition for the print the statement. **Addition will require you to use str() on the variables since they are integers!**

**3.**

After the print statements, declare the variable `tree_node`

that points to a Python dictionary with a key of `"data"`

pointing to `middle_value`

.

`tree_node`

represents the tree being created **in this function call**. We want a `tree_node`

created for each element in the list, so we’ll repeat this process on the left and right *sub-trees* using the appropriate half of the input list.

Now for the recursive calls!

Set the key of `"left_child"`

in `tree_node`

to be a recursive call to `build_bst()`

with the **left half of the list not including the middle value** as an argument.

Set the key of `"right_child"`

in `tree_node`

to be a recursive call to `build_bst()`

with the **right half of the list not including the middle value** as an argument.

It’s very important we don’t include the `middle_value`

in the lists we’re passing as arguments, or else we’ll create duplicate nodes!

Finally, return `tree_node`

.
As the recursive calls resolve and pop off the call stack, the final return value will be the *root* or “top” `tree_node`

which contains a reference to every other `tree_node`

.

**4.**

Congratulations! You’ve built up a recursive data structure with a recursive function!

This data structure can be used to find values in an efficient `O(logN)`

time.

Fill in the variable `runtime`

with the runtime of your `build_bst()`

function.

This runtime is a tricky one so don’t be afraid to use that hint!