# Treating Trees II: `treestep`

¶

Let’s consider a tree structure, similar to one described “Why functional programming matters” by John Hughes, that consists of a node value followed by zero or more child trees. (The asterisk is meant to indicate the Kleene star.)

```
tree = [] | [node tree*]
```

In the spirit of `step`

we are going to define a combinator
`treestep`

which expects a tree and three additional items: a
base-case function `[B]`

, and two quoted programs `[N]`

and `[C]`

.

```
tree [B] [N] [C] treestep
```

If the current tree node is empty then just execute `B`

:

```
[] [B] [N] [C] treestep
---------------------------
[] B
```

Otherwise, evaluate `N`

on the node value, `map`

the whole function
(abbreviated here as `K`

) over the child trees recursively, and then
combine the result with `C`

.

```
[node tree*] [B] [N] [C] treestep
--------------------------------------- w/ K == [B] [N] [C] treestep
node N [tree*] [K] map C
```

(Later on we’ll experiment with making `map`

part of `C`

so you can
use other combinators.)

## Derive the recursive function.¶

We can begin to derive it by finding the `ifte`

stage that `genrec`

will produce.

```
K == [not] [B] [R0] [R1] genrec
== [not] [B] [R0 [K] R1] ifte
```

So we just have to derive `J`

:

```
J == R0 [K] R1
```

The behavior of `J`

is to accept a (non-empty) tree node and arrive at
the desired outcome.

```
[node tree*] J
------------------------------
node N [tree*] [K] map C
```

So `J`

will have some form like:

```
J == ... [N] ... [K] ... [C] ...
```

Let’s dive in. First, unquote the node and `dip`

`N`

.

```
[node tree*] uncons [N] dip
node [tree*] [N] dip
node N [tree*]
```

Next, `map`

`K`

over the child trees and combine with `C`

.

```
node N [tree*] [K] map C
node N [tree*] [K] map C
node N [K.tree*] C
```

So:

```
J == uncons [N] dip [K] map C
```

Plug it in and convert to `genrec`

:

```
K == [not] [B] [J ] ifte
== [not] [B] [uncons [N] dip [K] map C] ifte
== [not] [B] [uncons [N] dip] [map C] genrec
```

## Extract the givens to parameterize the program.¶

Working backwards:

```
[not] [B] [uncons [N] dip] [map C] genrec
[B] [not] swap [uncons [N] dip] [map C] genrec
[B] [uncons [N] dip] [[not] swap] dip [map C] genrec
^^^^^^^^^^^^^^^^
[B] [[N] dip] [uncons] swoncat [[not] swap] dip [map C] genrec
[B] [N] [dip] cons [uncons] swoncat [[not] swap] dip [map C] genrec
^^^^^^^^^^^^^^^^^^^^^^^^^^^
```

Extract a couple of auxiliary definitions:

```
TS.0 == [[not] swap] dip
TS.1 == [dip] cons [uncons] swoncat
```

```
[B] [N] TS.1 TS.0 [map C] genrec
[B] [N] [map C] [TS.1 TS.0] dip genrec
[B] [N] [C] [map] swoncat [TS.1 TS.0] dip genrec
```

The givens are all to the left so we have our definition.

### (alternate) Extract the givens to parameterize the program.¶

Working backwards:

```
[not] [B] [uncons [N] dip] [map C] genrec
[not] [B] [N] [dip] cons [uncons] swoncat [map C] genrec
[B] [N] [not] roll> [dip] cons [uncons] swoncat [map C] genrec
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
```

## Define `treestep`

¶

```
from notebook_preamble import D, J, V, define, DefinitionWrapper
```

```
DefinitionWrapper.add_definitions('''
_treestep_0 == [[not] swap] dip
_treestep_1 == [dip] cons [uncons] swoncat
treegrind == [_treestep_1 _treestep_0] dip genrec
treestep == [map] swoncat treegrind
''', D)
```

## Examples¶

Consider trees, the nodes of which are integers. We can find the sum of all nodes in a tree with this function:

```
sumtree == [pop 0] [] [sum +] treestep
```

```
define('sumtree == [pop 0] [] [sum +] treestep')
```

Running this function on an empty tree value gives zero:

```
[] [pop 0] [] [sum +] treestep
------------------------------------
0
```

```
J('[] sumtree') # Empty tree.
```

```
0
```

Running it on a non-empty node:

```
[n tree*] [pop 0] [] [sum +] treestep
n [tree*] [[pop 0] [] [sum +] treestep] map sum +
n [ ... ] sum +
n m +
n+m
```

```
J('[23] sumtree') # No child trees.
```

```
23
```

```
J('[23 []] sumtree') # Child tree, empty.
```

```
23
```

```
J('[23 [2 [4]] [3]] sumtree') # Non-empty child trees.
```

```
32
```

```
J('[23 [2 [8] [9]] [3] [4 []]] sumtree') # Etc...
```

```
49
```

```
J('[23 [2 [8] [9]] [3] [4 []]] [pop 0] [] [cons sum] treestep') # Alternate "spelling".
```

```
49
```

```
J('[23 [2 [8] [9]] [3] [4 []]] [] [pop 23] [cons] treestep') # Replace each node.
```

```
[23 [23 [23] [23]] [23] [23 []]]
```

```
J('[23 [2 [8] [9]] [3] [4 []]] [] [pop 1] [cons] treestep')
```

```
[1 [1 [1] [1]] [1] [1 []]]
```

```
J('[23 [2 [8] [9]] [3] [4 []]] [] [pop 1] [cons] treestep sumtree')
```

```
6
```

```
J('[23 [2 [8] [9]] [3] [4 []]] [pop 0] [pop 1] [sum +] treestep') # Combine replace and sum into one function.
```

```
6
```

```
J('[4 [3 [] [7]]] [pop 0] [pop 1] [sum +] treestep') # Combine replace and sum into one function.
```

```
3
```

## Redefining the Ordered Binary Tree in terms of `treestep`

.¶

```
Tree = [] | [[key value] left right]
```

What kind of functions can we write for this with our `treestep`

?

The pattern for processing a non-empty node is:

```
node N [tree*] [K] map C
```

Plugging in our BTree structure:

```
[key value] N [left right] [K] map C
```

### Traversal¶

```
[key value] first [left right] [K] map i
key [value] [left right] [K] map i
key [left right] [K] map i
key [lkey rkey ] i
key lkey rkey
```

This doesn’t quite work:

```
J('[[3 0] [[2 0] [][]] [[9 0] [[5 0] [[4 0] [][]] [[8 0] [[6 0] [] [[7 0] [][]]][]]][]]] ["B"] [first] [i] treestep')
```

```
3 'B' 'B'
```

Doesn’t work because `map`

extracts the `first`

item of whatever its
mapped function produces. We have to return a list, rather than
depositing our results directly on the stack.

```
[key value] N [left right] [K] map C
[key value] first [left right] [K] map flatten cons
key [left right] [K] map flatten cons
key [[lk] [rk] ] flatten cons
key [ lk rk ] cons
[key lk rk ]
```

So:

```
[] [first] [flatten cons] treestep
```

```
J('[[3 0] [[2 0] [] []] [[9 0] [[5 0] [[4 0] [] []] [[8 0] [[6 0] [] [[7 0] [] []]] []]] []]] [] [first] [flatten cons] treestep')
```

```
[3 2 9 5 4 8 6 7]
```

There we go.

### In-order traversal¶

From here:

```
key [[lk] [rk]] C
key [[lk] [rk]] i
key [lk] [rk] roll<
[lk] [rk] key swons concat
[lk] [key rk] concat
[lk key rk]
```

So:

```
[] [i roll< swons concat] [first] treestep
```

```
J('[[3 0] [[2 0] [] []] [[9 0] [[5 0] [[4 0] [] []] [[8 0] [[6 0] [] [[7 0] [] []]] []]] []]] [] [uncons pop] [i roll< swons concat] treestep')
```

```
[2 3 4 5 6 7 8 9]
```

## With `treegrind`

?¶

The `treegrind`

function doesn’t include the `map`

combinator, so
the `[C]`

function must arrange to use some combinator on the quoted
recursive copy `[K]`

. With this function, the pattern for processing a
non-empty node is:

```
node N [tree*] [K] C
```

Plugging in our BTree structure:

```
[key value] N [left right] [K] C
```

```
J('[["key" "value"] ["left"] ["right"] ] ["B"] ["N"] ["C"] treegrind')
```

```
['key' 'value'] 'N' [['left'] ['right']] [[not] ['B'] [uncons ['N'] dip] ['C'] genrec] 'C'
```

`treegrind`

with `step`

¶

Iteration through the nodes

```
J('[[3 0] [[2 0] [] []] [[9 0] [[5 0] [[4 0] [] []] [[8 0] [[6 0] [] [[7 0] [] []]] []]] []]] [pop] ["N"] [step] treegrind')
```

```
[3 0] 'N' [2 0] 'N' [9 0] 'N' [5 0] 'N' [4 0] 'N' [8 0] 'N' [6 0] 'N' [7 0] 'N'
```

Sum the nodes’ keys.

```
J('0 [[3 0] [[2 0] [] []] [[9 0] [[5 0] [[4 0] [] []] [[8 0] [[6 0] [] [[7 0] [] []]] []]] []]] [pop] [first +] [step] treegrind')
```

```
44
```

Rebuild the tree using `map`

(imitating `treestep`

.)

```
J('[[3 0] [[2 0] [] []] [[9 0] [[5 0] [[4 0] [] []] [[8 0] [[6 0] [] [[7 0] [] []]] []]] []]] [] [[100 +] infra] [map cons] treegrind')
```

```
[[103 0] [[102 0] [] []] [[109 0] [[105 0] [[104 0] [] []] [[108 0] [[106 0] [] [[107 0] [] []]] []]] []]]
```

## Do we have the flexibility to reimplement `Tree-get`

?¶

I think we do:

```
[B] [N] [C] treegrind
```

We’ll start by saying that the base-case (the key is not in the tree) is user defined, and the per-node function is just the query key literal:

```
[B] [query_key] [C] treegrind
```

This means we just have to define `C`

from:

```
[key value] query_key [left right] [K] C
```

Let’s try `cmp`

:

```
C == P [T>] [E] [T<] cmp
[key value] query_key [left right] [K] P [T>] [E] [T<] cmp
```

### The predicate `P`

¶

Seems pretty easy (we must preserve the value in case the keys are equal):

```
[key value] query_key [left right] [K] P
[key value] query_key [left right] [K] roll<
[key value] [left right] [K] query_key [roll< uncons swap] dip
[key value] [left right] [K] roll< uncons swap query_key
[left right] [K] [key value] uncons swap query_key
[left right] [K] key [value] swap query_key
[left right] [K] [value] key query_key
P == roll< [roll< uncons swap] dip
```

(Possibly with a swap at the end? Or just swap `T<`

and `T>`

.)

So now:

```
[left right] [K] [value] key query_key [T>] [E] [T<] cmp
```

Becomes one of these three:

```
[left right] [K] [value] T>
[left right] [K] [value] E
[left right] [K] [value] T<
```

`T<`

and `T>`

¶

```
T< == pop [first] dip i
T> == pop [second] dip i
```

## Putting it together¶

```
T> == pop [first] dip i
T< == pop [second] dip i
E == roll> popop first
P == roll< [roll< uncons swap] dip
Tree-get == [P [T>] [E] [T<] cmp] treegrind
```

To me, that seems simpler than the `genrec`

version.

```
DefinitionWrapper.add_definitions('''
T> == pop [first] dip i
T< == pop [second] dip i
E == roll> popop first
P == roll< [roll< uncons swap] dip
Tree-get == [P [T>] [E] [T<] cmp] treegrind
''', D)
```

```
J('''\
[[3 13] [[2 12] [] []] [[9 19] [[5 15] [[4 14] [] []] [[8 18] [[6 16] [] [[7 17] [] []]] []]] []]]
[] [5] Tree-get
''')
```

```
15
```

```
J('''\
[[3 13] [[2 12] [] []] [[9 19] [[5 15] [[4 14] [] []] [[8 18] [[6 16] [] [[7 17] [] []]] []]] []]]
[pop "nope"] [25] Tree-get
''')
```

```
'nope'
```