This post is the fifth in a series.

In the previous post, we spent some time understanding folds for specific domain types.

In this post, we’ll broaden our horizons and look at how to use generic recursive types.

Here’s the contents of this series:

**Part 1: Introduction to recursive types and catamorphisms****Part 2: Catamorphism examples****Part 3: Introducing folds****Part 4: Understanding folds****Part 5: Generic recursive types****Part 6: Trees in the real world**- Defining a generic Tree type
- The Tree type in the real world
- Mapping the Tree type
- Example: Creating a directory listing
- Example: A parallel grep
- Example: Storing the file system in a database
- Example: Serializing a Tree to JSON
- Example: Deserializing a Tree from JSON
- Example: Deserializing a Tree from JSON - with error handling

Here’s a question: if you only have algebraic types, and you can only combine them as products (tuples, records) or sums (discriminated unions), then how can you make a list type just by using these operations?

The answer is, of course, recursion!

Let’s start with the most basic recursive type: the list.

I’m going to call my version `LinkedList`

, but it is basically the same as the `list`

type in F#.

So, how do you define a list in a recursive way?

Well, it’s either empty, or it consists of an element plus another list. In other words we can define it as a choice type (“discriminated union”) like this:

```
type LinkedList<'a> =
| Empty
| Cons of head:'a * tail:LinkedList<'a>
```

The `Empty`

case represents an empty list. The `Cons`

case has a tuple: the head element, and the tail, which is another list.

We can then define a particular `LinkedList`

value like this:

```
let linkedList = Cons (1, Cons (2, Cons(3, Empty)))
```

Using the native F# list type, the equivalent definition would be:

```
let linkedList = 1 :: 2 :: 3 :: []
```

which is just `[1; 2; 3]`

`cata`

for LinkedListFollowing the rules in the first post in this series,
we can mechanically create a `cata`

function by replacing `Empty`

and `Cons`

with `fEmpty`

and `fCons`

:

```
module LinkedList =
let rec cata fCons fEmpty list :'r=
let recurse = cata fCons fEmpty
match list with
| Empty ->
fEmpty
| Cons (element,list) ->
fCons element (recurse list)
```

*Note: We will be putting all the functions associated with LinkedList<'a> in a module called LinkedList. One nice thing about using generic types is that the type name does not clash with a similar module name!*

As always, the signatures of the case handling functions are parallel to the signatures of the type constructors, with `LinkedList`

replaced by `'r`

.

```
val cata :
fCons:('a -> 'r -> 'r) ->
fEmpty:'r ->
list:LinkedList<'a>
-> 'r
```

`fold`

for LinkedListWe can also create a top-down iterative `fold`

function using the rules in the earlier post.

```
module LinkedList =
let rec cata ...
let rec foldWithEmpty fCons fEmpty acc list :'r=
let recurse = foldWithEmpty fCons fEmpty
match list with
| Empty ->
fEmpty acc
| Cons (element,list) ->
let newAcc = fCons acc element
recurse newAcc list
```

This `foldWithEmpty`

function is not quite the same as the standard `List.fold`

function, because it has an extra function parameter for the empty case (`fEmpty`

).
However, if we eliminate that parameter and just return the accumulator we get this variant:

```
module LinkedList =
let rec fold fCons acc list :'r=
let recurse = fold fCons
match list with
| Empty ->
acc
| Cons (element,list) ->
let newAcc = fCons acc element
recurse newAcc list
```

If we compare the signature with the List.fold documentation we can see that they are equivalent,
with `'State`

replaced by `'r`

and `'T list`

replaced by `LinkedList<'a>`

:

```
LinkedList.fold : ('r -> 'a -> 'r ) -> 'r -> LinkedList<'a> -> 'r
List.fold : ('State -> 'T -> 'State) -> 'State -> 'T list -> 'State
```

Let’s test that `fold`

works by doing a small sum:

```
let linkedList = Cons (1, Cons (2, Cons(3, Empty)))
linkedList |> LinkedList.fold (+) 0
// Result => 6
```

`foldBack`

for LinkedListFinally we can create a `foldBack`

function, using the “function accumulator” approach described in the previous post:

```
module LinkedList =
let rec cata ...
let rec fold ...
let foldBack fCons list acc :'r=
let fEmpty' generator =
generator acc
let fCons' generator element=
fun innerResult ->
let newResult = fCons element innerResult
generator newResult
let initialGenerator = id
foldWithEmpty fCons' fEmpty' initialGenerator list
```

Again, if we compare the signature with the List.foldBack documentation, they are also equivalent,
with `'State`

replaced by `'r`

and `'T list`

replaced by `LinkedList<'a>`

:

```
LinkedList.foldBack : ('a -> 'r -> 'r ) -> LinkedList<'a> -> 'r -> 'r
List.foldBack : ('T -> 'State -> 'State) -> 'T list -> 'State -> 'State
```

`foldBack`

to convert between list typesIn the first post we noted that catamorphisms could be used for converting between types of similar structure.

Let’s demonstrate that now by creating some functions that convert from `LinkedList`

to the native `list`

type and back again.

To convert a `LinkedList`

to a native `list`

all we need to do is replace `Cons`

with `::`

and `Empty`

with `[]`

:

```
module LinkedList =
let toList linkedList =
let fCons head tail = head::tail
let initialState = []
foldBack fCons linkedList initialState
```

To convert the other way, we need to replace `::`

with `Cons`

and `[]`

with `Empty`

:

```
module LinkedList =
let ofList list =
let fCons head tail = Cons(head,tail)
let initialState = Empty
List.foldBack fCons list initialState
```

Simple! Let’s test `toList`

:

```
let linkedList = Cons (1, Cons (2, Cons(3, Empty)))
linkedList |> LinkedList.toList
// Result => [1; 2; 3]
```

and `ofList`

:

```
let list = [1;2;3]
list |> LinkedList.ofList
// Result => Cons (1,Cons (2,Cons (3,Empty)))
```

Both work as expected.

`foldBack`

to implement other functionsI said earlier that a catamorphism function (for linear lists, `foldBack`

) is the most basic function available for a recursive type, and in fact is the *only* function you need!

Let’s see for ourselves by implementing some other common functions using `foldBack`

.

Here’s `map`

defined in terms of `foldBack`

:

```
module LinkedList =
/// map a function "f" over all elements
let map f list =
// helper function
let folder head tail =
Cons(f head,tail)
foldBack folder list Empty
```

And here’s a test:

```
let linkedList = Cons (1, Cons (2, Cons(3, Empty)))
linkedList |> LinkedList.map (fun i -> i+10)
// Result => Cons (11,Cons (12,Cons (13,Empty)))
```

Here’s `filter`

defined in terms of `foldBack`

:

```
module LinkedList =
/// return a new list of elements for which "pred" is true
let filter pred list =
// helper function
let folder head tail =
if pred head then
Cons(head,tail)
else
tail
foldBack folder list Empty
```

And here’s a test:

```
let isOdd n = (n%2=1)
let linkedList = Cons (1, Cons (2, Cons(3, Empty)))
linkedList |> LinkedList.filter isOdd
// Result => Cons (1,Cons (3,Empty))
```

Finally, here’s `rev`

defined in terms of `fold`

:

```
/// reverse the elements of the list
let rev list =
// helper function
let folder tail head =
Cons(head,tail)
fold folder Empty list
```

And here’s a test:

```
let linkedList = Cons (1, Cons (2, Cons(3, Empty)))
linkedList |> LinkedList.rev
// Result => Cons (3,Cons (2,Cons (1,Empty)))
```

So, I hope you’re convinced!

I mentioned earlier that there was an alternative and (sometimes) more efficient way to implement `foldBack`

without using generators or continuations.

As we have seen, `foldBack`

is reverse iteration, which means that it is the same as `fold`

applied to a reversed list!

So we could implement it like this:

```
let foldBack_ViaRev fCons list acc :'r=
let fCons' acc element =
// just swap the params!
fCons element acc
list
|> rev
|> fold fCons' acc
```

It involves making an extra copy of the list, but on the other hand there is no longer a large set of pending continuations. It might be worth comparing the profile of the two versions in your environment if performance is an issue.

In the rest of this post, we’ll look at the `Gift`

type and see if we can make it more generic.

As a reminder, here is the original design:

```
type Gift =
| Book of Book
| Chocolate of Chocolate
| Wrapped of Gift * WrappingPaperStyle
| Boxed of Gift
| WithACard of Gift * message:string
```

Three of the cases are recursive and two are non-recursive.

Now, the focus of this particular design was on modelling the domain, which is why there are so many separate cases.

But if we want to focus on *reusability* instead of domain modelling, then we should simplify the design to the essentials, and all these special cases now become a hindrance.

To make this ready for reuse, then, let’s collapse all the non-recursive cases into one case, say `GiftContents`

,
and all the recursive cases into another case, say `GiftDecoration`

, like this:

```
// unified data for non-recursive cases
type GiftContents =
| Book of Book
| Chocolate of Chocolate
// unified data for recursive cases
type GiftDecoration =
| Wrapped of WrappingPaperStyle
| Boxed
| WithACard of string
type Gift =
// non-recursive case
| Contents of GiftContents
// recursive case
| Decoration of Gift * GiftDecoration
```

The main `Gift`

type has only two cases now: the non-recursive one and the recursive one.

Now that the type is simplified, we can “genericize” it by allowing *any* kind of contents *and* any kind of decoration.

```
type Container<'ContentData,'DecorationData> =
| Contents of 'ContentData
| Decoration of 'DecorationData * Container<'ContentData,'DecorationData>
```

And as before, we can mechanically create a `cata`

and `fold`

and `foldBack`

for it, using the standard process:

```
module Container =
let rec cata fContents fDecoration (container:Container<'ContentData,'DecorationData>) :'r =
let recurse = cata fContents fDecoration
match container with
| Contents contentData ->
fContents contentData
| Decoration (decorationData,subContainer) ->
fDecoration decorationData (recurse subContainer)
(*
val cata :
// function parameters
fContents:('ContentData -> 'r) ->
fDecoration:('DecorationData -> 'r -> 'r) ->
// input
container:Container<'ContentData,'DecorationData> ->
// return value
'r
*)
let rec fold fContents fDecoration acc (container:Container<'ContentData,'DecorationData>) :'r =
let recurse = fold fContents fDecoration
match container with
| Contents contentData ->
fContents acc contentData
| Decoration (decorationData,subContainer) ->
let newAcc = fDecoration acc decorationData
recurse newAcc subContainer
(*
val fold :
// function parameters
fContents:('a -> 'ContentData -> 'r) ->
fDecoration:('a -> 'DecorationData -> 'a) ->
// accumulator
acc:'a ->
// input
container:Container<'ContentData,'DecorationData> ->
// return value
'r
*)
let foldBack fContents fDecoration (container:Container<'ContentData,'DecorationData>) :'r =
let fContents' generator contentData =
generator (fContents contentData)
let fDecoration' generator decorationData =
let newGenerator innerValue =
let newInnerValue = fDecoration decorationData innerValue
generator newInnerValue
newGenerator
fold fContents' fDecoration' id container
(*
val foldBack :
// function parameters
fContents:('ContentData -> 'r) ->
fDecoration:('DecorationData -> 'r -> 'r) ->
// input
container:Container<'ContentData,'DecorationData> ->
// return value
'r
*)
```

Let’s convert the gift type to this generic Container type:

```
type Gift = Container<GiftContents,GiftDecoration>
```

Now we need some helper methods to construct values while hiding the “real” cases of the generic type:

```
let fromBook book =
Contents (Book book)
let fromChoc choc =
Contents (Chocolate choc)
let wrapInPaper paperStyle innerGift =
let container = Wrapped paperStyle
Decoration (container, innerGift)
let putInBox innerGift =
let container = Boxed
Decoration (container, innerGift)
let withCard message innerGift =
let container = WithACard message
Decoration (container, innerGift)
```

Finally we can create some test values:

```
let wolfHall = {title="Wolf Hall"; price=20m}
let yummyChoc = {chocType=SeventyPercent; price=5m}
let birthdayPresent =
wolfHall
|> fromBook
|> wrapInPaper HappyBirthday
|> withCard "Happy Birthday"
let christmasPresent =
yummyChoc
|> fromChoc
|> putInBox
|> wrapInPaper HappyHolidays
```

`totalCost`

function using the Container typeThe “total cost” function can be written using `fold`

, since it doesn’t need any inner data.

Unlike the earlier implementations, we only have two function parameters, `fContents`

and `fDecoration`

, so each of these
will need some pattern matching to get at the “real” data.

Here’s the code:

```
let totalCost gift =
let fContents costSoFar contentData =
match contentData with
| Book book ->
costSoFar + book.price
| Chocolate choc ->
costSoFar + choc.price
let fDecoration costSoFar decorationInfo =
match decorationInfo with
| Wrapped style ->
costSoFar + 0.5m
| Boxed ->
costSoFar + 1.0m
| WithACard message ->
costSoFar + 2.0m
// initial accumulator
let initialAcc = 0m
// call the fold
Container.fold fContents fDecoration initialAcc gift
```

And the code works as expected:

```
birthdayPresent |> totalCost
// 22.5m
christmasPresent |> totalCost
// 6.5m
```

`description`

function using the Container typeThe “description” function needs to be written using `foldBack`

, since it *does* need the inner data. As with the code above,
we need some pattern matching to get at the “real” data for each case.

```
let description gift =
let fContents contentData =
match contentData with
| Book book ->
sprintf "'%s'" book.title
| Chocolate choc ->
sprintf "%A chocolate" choc.chocType
let fDecoration decorationInfo innerText =
match decorationInfo with
| Wrapped style ->
sprintf "%s wrapped in %A paper" innerText style
| Boxed ->
sprintf "%s in a box" innerText
| WithACard message ->
sprintf "%s with a card saying '%s'" innerText message
// main call
Container.foldBack fContents fDecoration gift
```

And again the code works as we want:

```
birthdayPresent |> description
// CORRECT "'Wolf Hall' wrapped in HappyBirthday paper with a card saying 'Happy Birthday'"
christmasPresent |> description
// CORRECT "SeventyPercent chocolate in a box wrapped in HappyHolidays paper"
```

That all looks quite nice, doesn’t it?

But I have to confess that I have been holding something back.

None of that code above was strictly necessary, because it turns out that there is yet *another* way to model a `Gift`

,
without creating any new generic types at all!

The `Gift`

type is basically a linear sequence of decorations, with some content as the final step. We can just model this as a pair – a `Content`

and a list of `Decoration`

.
Or to make it a little friendlier, a record with two fields: one for the content and one for the decorations.

```
type Gift = {contents: GiftContents; decorations: GiftDecoration list}
```

That’s it! No other new types needed!

As before, let’s create some helpers to construct values using this type:

```
let fromBook book =
{ contents = (Book book); decorations = [] }
let fromChoc choc =
{ contents = (Chocolate choc); decorations = [] }
let wrapInPaper paperStyle innerGift =
let decoration = Wrapped paperStyle
{ innerGift with decorations = decoration::innerGift.decorations }
let putInBox innerGift =
let decoration = Boxed
{ innerGift with decorations = decoration::innerGift.decorations }
let withCard message innerGift =
let decoration = WithACard message
{ innerGift with decorations = decoration::innerGift.decorations }
```

With these helper functions, the way the values are constructed is *identical* to the previous version. This is why it is good to hide your raw constructors, folks!

```
let wolfHall = {title="Wolf Hall"; price=20m}
let yummyChoc = {chocType=SeventyPercent; price=5m}
let birthdayPresent =
wolfHall
|> fromBook
|> wrapInPaper HappyBirthday
|> withCard "Happy Birthday"
let christmasPresent =
yummyChoc
|> fromChoc
|> putInBox
|> wrapInPaper HappyHolidays
```

`totalCost`

function using the record typeThe `totalCost`

function is even easier to write now.

```
let totalCost gift =
let contentCost =
match gift.contents with
| Book book ->
book.price
| Chocolate choc ->
choc.price
let decorationFolder costSoFar decorationInfo =
match decorationInfo with
| Wrapped style ->
costSoFar + 0.5m
| Boxed ->
costSoFar + 1.0m
| WithACard message ->
costSoFar + 2.0m
let decorationCost =
gift.decorations |> List.fold decorationFolder 0m
// total cost
contentCost + decorationCost
```

`description`

function using the record typeSimilarly, the `description`

function is also easy to write.

```
let description gift =
let contentDescription =
match gift.contents with
| Book book ->
sprintf "'%s'" book.title
| Chocolate choc ->
sprintf "%A chocolate" choc.chocType
let decorationFolder decorationInfo innerText =
match decorationInfo with
| Wrapped style ->
sprintf "%s wrapped in %A paper" innerText style
| Boxed ->
sprintf "%s in a box" innerText
| WithACard message ->
sprintf "%s with a card saying '%s'" innerText message
List.foldBack decorationFolder gift.decorations contentDescription
```

If you are confused by this plethora of designs, I don’t blame you!

But as it happens, the three different definitions are actually interchangable:

**The original version**

```
type Gift =
| Book of Book
| Chocolate of Chocolate
| Wrapped of Gift * WrappingPaperStyle
| Boxed of Gift
| WithACard of Gift * message:string
```

**The generic container version**

```
type Container<'ContentData,'DecorationData> =
| Contents of 'ContentData
| Decoration of 'DecorationData * Container<'ContentData,'DecorationData>
type GiftContents =
| Book of Book
| Chocolate of Chocolate
type GiftDecoration =
| Wrapped of WrappingPaperStyle
| Boxed
| WithACard of string
type Gift = Container<GiftContents,GiftDecoration>
```

**The record version**

```
type GiftContents =
| Book of Book
| Chocolate of Chocolate
type GiftDecoration =
| Wrapped of WrappingPaperStyle
| Boxed
| WithACard of string
type Gift = {contents: GiftContents; decorations: GiftDecoration list}
```

If this is not obvious, it might be helpful to read my post on data type sizes. It explains how two types can be “equivalent”, even though they appear to be completely different at first glance.

So which design is best? The answer is, as always, “it depends”.

For modelling and documenting a domain, I like the first design with the five explicit cases. Being easy for other people to understand is more important to me than introducing abstraction for the sake of reusability.

If I wanted a reusable model that was applicable in many situations, I’d probably choose the second (“Container”) design. It seems to me that this type does represent a commonly encountered situation, where the contents are one kind of thing and the wrappers are another kind of thing. This abstraction is therefore likely to get some use.

The final “pair” model is fine, but by separating the two components, we’ve over-abstracted the design for this scenario. In other situations, this design might be a great fit (e.g. the decorator pattern), but not here, in my opinion.

There is one further choice which gives you the best of all worlds.

As I noted above, all the designs are logically equivalent, which means there are “lossless” mappings between them. In that case, your “public” design can be the domain-oriented one, like the first one, but behind the scenes you can map it to a more efficient and reusable “private” type.

Even the F# list implementation itself does this.
For example, some of the functions in the `List`

module, such `foldBack`

and `sort`

, convert the list into an array, do the operations, and then convert it back to a list again.

In this post we looked at some ways of modelling the `Gift`

as a generic type, and the pros and cons of each approach.

In the next post we’ll look at real-world examples of using a generic recursive type.

*The source code for this post is available at this gist.*