An Introduction to Elm Series: Solution to ‘Binary Tree’ example
In http://elm-lang.org/examples/binary-tree, we are given a basic implementation of a binary tree and we are asked to extend it in various ways.
To implement the Breadth First Traversal, we had to use another module that acts like a Queue. The code of that Queue is available at this post http://bytefreaks.net/programming-2/elm/an-introduction-to-elm-series-solution-to-binary-tree-example-supplementary-module-queue and it is also included in the zip file with the code of this solution ( [download id=”1828″]).
Below you will find our solutions to these challenges.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 | {- OVERVIEW ------------------------------------------------------ A "Tree" represents a binary tree. A "Node" in a binary tree always has two children. A tree can also be "Empty". Below I have defined "Tree" and a number of useful functions. This example also includes some challenge problems! -----------------------------------------------------------------}module Main exposing (..)import Queue exposing (Queue)import Html exposing (Html, div, text, br)import Html.Attributes exposing (style)-- TREEStype Tree a = Empty | Node a (Tree a) (Tree a)empty : Tree aempty = Emptysingleton : a -> Tree asingleton v = Node v Empty Emptyinsert : comparable -> Tree comparable -> Tree comparableinsert x tree = case tree of Empty -> singleton x Node y left right -> if x > y then Node y left (insert x right) else if x < y then Node y (insert x left) right else treefromList : List comparable -> Tree comparablefromList xs = List.foldl insert empty xsdepth : Tree a -> Intdepth tree = case tree of Empty -> 0 Node v left right -> 1 + max (depth left) (depth right)map : (a -> b) -> Tree a -> Tree bmap f tree = case tree of Empty -> Empty Node v left right -> Node (f v) (map f left) (map f right)-- (1) Sum all of the elements of a tree.sum : Tree number -> numbersum tree = case tree of Empty -> 0 Node value left right -> value + sum left + sum right-- (2) Flatten a tree into a list.flatten : Tree a -> List aflatten tree = case tree of Empty -> [] Node value left right -> flatten left ++ [ value ] ++ flatten right-- (3) Check to see if an element is in a given tree.isElement : a -> Tree a -> BoolisElement element tree = case tree of Empty -> False Node value left right -> -- We short circuit the evaluation, if we find early that the element is in the tree we stop the recursion down this path. if value == element then True else isElement element left || isElement element right{--(4) Write a general fold function that acts on trees. The fold function does not need to guarantee a particular order of traversal.--}fold : (a -> b -> b) -> b -> Tree a -> bfold function element tree = case tree of Node value left right -> function value (fold function (fold function element right) left) Empty -> element{--(5) Use "fold" to do exercises 1-3 in one line each. The best readable versions I have come up have the following length in characters including spaces and function name: sum: 16 flatten: 21 isElement: 46 See if you can match or beat me! Don't forget about currying and partial application!--}-- The following functions are called as parameters for the fold function-- Sum the values of all nodes in the treesumByFold : number -> number -> numbersumByFold elementA elementB = elementA + elementB-- Flatten the tree in a pre-order fashionflattenByFold : a -> List a -> List aflattenByFold element list = [ element ] ++ list-- Count the total number of nodes in the treecountByFold : number -> number -> numbercountByFold element count' = 1 + count'-- The following functions are using fold internally-- Sum the values of all nodes in the treesumUsingFold : Tree number -> numbersumUsingFold = fold (+) 0-- Flatten the tree in an in-order fashionflattenUsingFold : Tree a -> List aflattenUsingFold = fold (::) []-- Checks if a certain element is in the treeisElementUsingFold : a -> Tree a -> BoolisElementUsingFold element = fold ((==) element >> (||)) False-- (6) Can "fold" be used to implement "map" or "depth"?-- TODO{--(7) Try experimenting with different ways to traverse a tree: pre-order, in-order, post-order, depth-first, etc. More info at: http://en.wikipedia.org/wiki/Tree_traversal--}-- Depth-firstpreOrder : Tree a -> List apreOrder tree = case tree of Empty -> [] Node value left right -> [ value ] ++ preOrder left ++ preOrder rightinOrder : Tree a -> List ainOrder tree = case tree of Empty -> [] Node value left right -> inOrder left ++ [ value ] ++ inOrder rightpostOrder : Tree a -> List apostOrder tree = case tree of Empty -> [] Node value left right -> postOrder left ++ postOrder right ++ [ value ]-- Breadth-firstgo : Queue (Tree a) -> List ago queue = let ( maybe', queue' ) = Queue.dequeue queue in case maybe' of Just node -> case node of Empty -> go queue' Node value left right -> [ value ] ++ go (Queue.enqueue right (Queue.enqueue left queue')) Nothing -> []levelOrder : Tree a -> List alevelOrder tree = case tree of Empty -> [] Node value left right -> go (Queue.enqueue tree (Queue.init))-- PLAYGROUNDdeepTree = fromList [ 1, 2, 3 ]niceTree = fromList [ 2, 1, 3 ]deepTree' = fromList [ 1, 2, 3, 4, 5, 6, 7 ]niceTree' = fromList [ 4, 6, 2, 3, 5, 1, 7 ]main = div [ style [ ( "font-family", "monospace" ) ] ] [ display "depth deepTree" (depth deepTree) , display "depth niceTree" (depth niceTree) , display "incremented" (map (\n -> n + 1) niceTree) , display "sum deepTree" (sum deepTree) , display "sum niceTree" (sum niceTree) , display "fold sumByFold 0 deepTree" (fold sumByFold 0 deepTree) , display "fold sumByFold 0 niceTree" (fold sumByFold 0 niceTree) , display "sumUsingFold deepTree" (sumUsingFold deepTree) , display "sumUsingFold niceTree" (sumUsingFold niceTree) , display "flatten deepTree" (flatten deepTree) , display "flatten niceTree" (flatten niceTree) , display "fold flattenByFold deepTree" (fold flattenByFold [] deepTree) , display "fold flattenByFold niceTree" (fold flattenByFold [] niceTree) , display "flattenUsingFold deepTree" (flattenUsingFold deepTree) , display "flattenUsingFold niceTree" (flattenUsingFold niceTree) , display "isElement deepTree 3" (isElement 3 deepTree) , display "isElement deepTree 4" (isElement 4 deepTree) , display "isElement niceTree 3" (isElement 3 niceTree) , display "isElement niceTree 4" (isElement 4 niceTree) , display "isElementUsingFold 3 deepTree" (isElementUsingFold 3 deepTree) , display "isElementUsingFold 4 deepTree" (isElementUsingFold 4 deepTree) , display "isElementUsingFold 3 niceTree" (isElementUsingFold 3 niceTree) , display "isElementUsingFold 4 niceTree" (isElementUsingFold 4 niceTree) , display "fold countByFold 0 deepTree" (fold countByFold 0 deepTree) , display "fold countByFold 0 niceTree" (fold countByFold 0 niceTree) , display "preOrder deepTree" (preOrder deepTree) , display "preOrder niceTree" (preOrder niceTree) , display "inOrder deepTree" (inOrder deepTree) , display "inOrder niceTree" (inOrder niceTree) , display "postOrder deepTree" (postOrder deepTree) , display "postOrder niceTree" (postOrder niceTree) , display "levelOrder deepTree" (levelOrder deepTree) , display "levelOrder niceTree" (levelOrder niceTree) , br [] [] , display "depth deepTree'" (depth deepTree') , display "depth niceTree'" (depth niceTree') , display "incremented" (map (\n -> n + 1) niceTree') , display "sum deepTree'" (sum deepTree') , display "sum niceTree'" (sum niceTree') , display "fold sumByFold 0 deepTree'" (fold sumByFold 0 deepTree') , display "fold sumByFold 0 niceTree'" (fold sumByFold 0 niceTree') , display "sumUsingFold deepTree'" (sumUsingFold deepTree') , display "sumUsingFold niceTree'" (sumUsingFold niceTree') , display "flatten deepTree'" (flatten deepTree') , display "flatten niceTree'" (flatten niceTree') , display "fold flattenByFold deepTree'" (fold flattenByFold [] deepTree') , display "fold flattenByFold niceTree'" (fold flattenByFold [] niceTree') , display "flattenUsingFold deepTree'" (flattenUsingFold deepTree') , display "flattenUsingFold niceTree'" (flattenUsingFold niceTree') , display "isElement deepTree' 3" (isElement 3 deepTree') , display "isElement deepTree' 4" (isElement 4 deepTree') , display "isElement niceTree' 3" (isElement 3 niceTree') , display "isElement niceTree' 4" (isElement 4 niceTree') , display "isElementUsingFold 3 deepTree'" (isElementUsingFold 3 deepTree') , display "isElementUsingFold 4 deepTree'" (isElementUsingFold 4 deepTree') , display "isElementUsingFold 3 niceTree'" (isElementUsingFold 3 niceTree') , display "isElementUsingFold 4 niceTree'" (isElementUsingFold 4 niceTree') , display "fold countByFold 0 deepTree'" (fold countByFold 0 deepTree') , display "fold countByFold 0 niceTree'" (fold countByFold 0 niceTree') , display "preOrder deepTree'" (preOrder deepTree') , display "preOrder niceTree'" (preOrder niceTree') , display "inOrder deepTree'" (inOrder deepTree') , display "inOrder niceTree'" (inOrder niceTree') , display "postOrder deepTree'" (postOrder deepTree') , display "postOrder niceTree'" (postOrder niceTree') , display "levelOrder deepTree'" (levelOrder deepTree') , display "levelOrder niceTree'" (levelOrder niceTree') ; ]display : String -> a -> Html msgdisplay name value = div [] [ text (name ++ " ==> " ++ toString value) ]{-----------------------------------------------------------------Exercises:(1) Sum all of the elements of a tree. sum : Tree number -> number(2) Flatten a tree into a list. flatten : Tree a -> List a(3) Check to see if an element is in a given tree. isElement : a -> Tree a -> Bool(4) Write a general fold function that acts on trees. The fold function does not need to guarantee a particular order of traversal. fold : (a -> b -> b) -> b -> Tree a -> b(5) Use "fold" to do exercises 1-3 in one line each. The best readable versions I have come up have the following length in characters including spaces and function name: sum: 16 flatten: 21 isElement: 46 See if you can match or beat me! Don't forget about currying and partial application!(6) Can "fold" be used to implement "map" or "depth"?(7) Try experimenting with different ways to traverse a tree: pre-order, in-order, post-order, depth-first, etc. More info at: http://en.wikipedia.org/wiki/Tree_traversal-----------------------------------------------------------------} |
The output of the above is the following
depth deepTree ==> 3 depth niceTree ==> 2 incremented ==> Node 3 (Node 2 Empty Empty) (Node 4 Empty Empty) sum deepTree ==> 6 sum niceTree ==> 6 fold sumByFold 0 deepTree ==> 6 fold sumByFold 0 niceTree ==> 6 sumUsingFold deepTree ==> 6 sumUsingFold niceTree ==> 6 flatten deepTree ==> [1,2,3] flatten niceTree ==> [1,2,3] fold flattenByFold deepTree ==> [1,2,3] fold flattenByFold niceTree ==> [2,1,3] flattenUsingFold deepTree ==> [1,2,3] flattenUsingFold niceTree ==> [2,1,3] isElement deepTree 3 ==> True isElement deepTree 4 ==> False isElement niceTree 3 ==> True isElement niceTree 4 ==> False isElementUsingFold 3 deepTree ==> True isElementUsingFold 4 deepTree ==> False isElementUsingFold 3 niceTree ==> True isElementUsingFold 4 niceTree ==> False fold countByFold 0 deepTree ==> 3 fold countByFold 0 niceTree ==> 3 preOrder deepTree ==> [1,2,3] preOrder niceTree ==> [2,1,3] inOrder deepTree ==> [1,2,3] inOrder niceTree ==> [1,2,3] postOrder deepTree ==> [3,2,1] postOrder niceTree ==> [1,3,2] levelOrder deepTree ==> [1,2,3] levelOrder niceTree ==> [2,1,3] depth deepTree' ==> 7 depth niceTree' ==> 3 incremented ==> Node 5 (Node 3 (Node 2 Empty Empty) (Node 4 Empty Empty)) (Node 7 (Node 6 Empty Empty) (Node 8 Empty Empty)) sum deepTree' ==> 28 sum niceTree' ==> 28 fold sumByFold 0 deepTree' ==> 28 fold sumByFold 0 niceTree' ==> 28 sumUsingFold deepTree' ==> 28 sumUsingFold niceTree' ==> 28 flatten deepTree' ==> [1,2,3,4,5,6,7] flatten niceTree' ==> [1,2,3,4,5,6,7] fold flattenByFold deepTree' ==> [1,2,3,4,5,6,7] fold flattenByFold niceTree' ==> [4,2,1,3,6,5,7] flattenUsingFold deepTree' ==> [1,2,3,4,5,6,7] flattenUsingFold niceTree' ==> [4,2,1,3,6,5,7] isElement deepTree' 3 ==> True isElement deepTree' 4 ==> True isElement niceTree' 3 ==> True isElement niceTree' 4 ==> True isElementUsingFold 3 deepTree' ==> True isElementUsingFold 4 deepTree' ==> True isElementUsingFold 3 niceTree' ==> True isElementUsingFold 4 niceTree' ==> True fold countByFold 0 deepTree' ==> 7 fold countByFold 0 niceTree' ==> 7 preOrder deepTree' ==> [1,2,3,4,5,6,7] preOrder niceTree' ==> [4,2,1,3,6,5,7] inOrder deepTree' ==> [1,2,3,4,5,6,7] inOrder niceTree' ==> [1,2,3,4,5,6,7] postOrder deepTree' ==> [7,6,5,4,3,2,1] postOrder niceTree' ==> [1,3,2,5,7,6,4] levelOrder deepTree' ==> [1,2,3,4,5,6,7] levelOrder niceTree' ==> [4,2,6,1,3,5,7]
You can download the solution from here [download id=”1828″]
