 Register  # 2.3. Functional Programming - Recurring On Lists That Aren't Flat¶

## 2.3.1. Deep Recursion in FP (1)¶

In the previous section, we limited our treatment of list-processing functions to operating on flat lists of integers, that is, list that do not themselves contain nested inner lists. In this section, we will consider how to operate on lists that can contain not only integers, but also lists. This will lead to a discussion of "deep recursion," which can handle trees represented as lists of lists of ... lists of integers nested arbitrarily deep.

A good guideline for deep recursion is the following. When recurring on a list of creatures lst that may themselves be lists, recur on both fp.hd(lst) and fp.tl(lst) – just make sure lst is not empty and that fp.hd(lst) is a list.

Consider the following tree_test list as an example:

var tree_test = [14, [7, [], [12, [], []]],
[26, [20, [17, [], []],
[] ],
[31, [], []]]]


We want to develop a function that takes in an integer tree represented as a list of lists of …lists of integers and returns the sum of the integers in the tree.

var sumTree = function (ns) {
if (fp.isNull(ns)) {
return 0;
} else if (???????) {
return ?????????;
} else {
return ?????????;
}
}


Once you have decided how to fill in the ??????? in the preceding example, consider the slight modification in the following review problem.

## 2.3.2. Deep Recursion 2¶

Notice that our tree_test is actually a binary search tree under the interpretation that the first nested list following a number is the left subtree of that number and the second nested list following a number is the right subtree. Using this represented of the tree, write a function path

var path = function (n, bst)

where n is a number and bst is a binary search tree that contains the number n. path should return a list of 0’s and 1’s showing how to find the node containing n, where 1 indicates "go right" and 0 indicates "go left". If n is found at the root, the empty list is returned. Example usage:

> var tree_test = [14, [7, [], [12, [], []]],
[26, [20, [17, [], []],
[] ],
[31, [], []]]]
> path(17, tree_test)
[1, 0, 0]


In writing this function, you should be using cons to construct the returned list as you progress through the tree using a deep recursion strategy. Once you feel you have a correct solution, use similar logic in developing a solution for the following review problem.

This problem again deals with deep recursion and more specifically with the subst function that was previously described.

## 2.3.3. Deep Recursion 3¶

This problem is similar to (and assumes that you have solved) the previous problem.

## 2.3.4. Deep Recursion 4¶

As a final example to give you a lot more practice with deep recursion, consider the following randomized exercise. You have to solve it correctly three times in a row.  