# Treating Trees II: `treestep` Let's consider a tree structure, similar to one described ["Why functional programming matters" by John Hughes](https://www.cs.kent.ac.uk/people/staff/dat/miranda/whyfp90.pdf), that consists of a node value followed by zero or more child trees. (The asterisk is meant to indicate the [Kleene star](https://en.wikipedia.org/wiki/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` ```python from notebook_preamble import D, J, V, define, DefinitionWrapper ``` ```python 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 ```python define('sumtree == [pop 0] [] [sum +] treestep') ``` Running this function on an empty tree value gives zero: [] [pop 0] [] [sum +] treestep ------------------------------------ 0 ```python 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 ```python J('[23] sumtree') # No child trees. ``` 23 ```python J('[23 []] sumtree') # Child tree, empty. ``` 23 ```python J('[23 [2 [4]] [3]] sumtree') # Non-empty child trees. ``` 32 ```python J('[23 [2 [8] [9]] [3] [4 []]] sumtree') # Etc... ``` 49 ```python J('[23 [2 [8] [9]] [3] [4 []]] [pop 0] [] [cons sum] treestep') # Alternate "spelling". ``` 49 ```python J('[23 [2 [8] [9]] [3] [4 []]] [] [pop 23] [cons] treestep') # Replace each node. ``` [23 [23 [23] [23]] [23] [23 []]] ```python J('[23 [2 [8] [9]] [3] [4 []]] [] [pop 1] [cons] treestep') ``` [1 [1 [1] [1]] [1] [1 []]] ```python J('[23 [2 [8] [9]] [3] [4 []]] [] [pop 1] [cons] treestep sumtree') ``` 6 ```python J('[23 [2 [8] [9]] [3] [4 []]] [pop 0] [pop 1] [sum +] treestep') # Combine replace and sum into one function. ``` 6 ```python 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: ```python 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 ```python 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 ```python 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 ```python 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 ```python 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. ```python 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`.) ```python 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< ### `E` Easy. E == roll> popop first ### `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. ```python 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) ``` ```python J('''\ [[3 13] [[2 12] [] []] [[9 19] [[5 15] [[4 14] [] []] [[8 18] [[6 16] [] [[7 17] [] []]] []]] []]] [] [5] Tree-get ''') ``` 15 ```python J('''\ [[3 13] [[2 12] [] []] [[9 19] [[5 15] [[4 14] [] []] [[8 18] [[6 16] [] [[7 17] [] []]] []]] []]] [pop "nope"] [25] Tree-get ''') ``` 'nope'