Complete Multiway Trees

Author: Abhijit Sarkar

README.md

@@ -138,15 +138,15 @@ P66 (***) Layout a binary tree (3).
 
 P70B Omitted; we can only create well-formed trees.
 
-P70C (*) Count the nodes of a multiway tree.
+[P70C](mtree/src/P70C.scala) (*) Count the nodes of a multiway tree.
 
-P70 (**) Tree construction from a node string.
+[P70](mtree/src/P70.scala) (**) Tree construction from a node string.
 
-P71 (*) Determine the internal path length of a tree.
+[P71](mtree/src/P71.scala) (*) Determine the internal path length of a tree.
 
-P72 (*) Construct the postorder sequence of the tree nodes.
+[P72](mtree/src/P72.scala) (*) Construct the postorder sequence of the tree nodes.
 
-P73 (**) Lisp-like tree representation.
+[P73](mtree/src/P73.scala) (**) Lisp-like tree representation.
 
 ## Graphs
 

bintree/src/DList.scala renamed to mtree/src/DList.scala

@@ -1,4 +1,4 @@
-package bintree
+package mtree
 
 final case class DList[A](run: List[A] => List[A]):
   // O(1)

mtree/src/MTree.scala

@@ -0,0 +1,5 @@
+package mtree
+
+final case class MTree[+T](value: T, children: Seq[MTree[T]] = Nil):
+  override def toString: String =
+    s"M(${value.toString} {${children.map(_.toString).mkString(",")}})"

mtree/src/P70.scala

@@ -0,0 +1,51 @@
+package mtree
+
+// P70 (**) Tree construction from a node string.
+//
+// We suppose that the nodes of a multiway tree contain single characters.
+// In the depth-first order sequence of its nodes, a special character ^
+// has been inserted whenever, during the tree traversal, the move is a
+// backtrack to the previous level.
+//
+// By this rule, the tree in the figure opposite is represented as:
+//
+// afg^^c^bd^e^^^
+// Define the syntax of the string and write a function string2MTree to construct
+// an MTree from a String.  Make the function an implicit conversion from String.
+// Write the reverse function, and make it the toString method of MTree.
+//
+// scala> MTree('a', List(MTree('f', List(MTree('g'))), MTree('c'), MTree('b', List(MTree('d'), MTree('e'))))).toString
+// res0: String = afg^^c^bd^e^^^
+object P70:
+  def string2MTree(s: String): MTree[Char] =
+    /*
+    First recursive dfs call collects the children,
+    second recursive dfs call collects the siblings.
+
+    x=g, xs=^^c^bd^e^^^, ys=^c^bd^e^^^, zs=c^bd^e^^^
+    x=e, xs=^^^, ys=^^, zs=^
+    x=d, xs=^e^^^, ys=e^^^, zs=^
+    x=b, xs=d^e^^^, ys=^, zs=
+    x=c, xs=^bd^e^^^, ys=bd^e^^^, zs=
+    x=f, xs=g^^c^bd^e^^^, ys=c^bd^e^^^, zs=
+    x=a, xs=fg^^c^bd^e^^^, ys=, zs=
+     */
+    def loop(s: String): (List[MTree[Char]], String) =
+      s match
+        case ""      => (Nil, "")
+        case s"^$xs" => (Nil, xs)
+        case _ =>
+          val (x, xs)        = (s.head, s.tail)
+          val (children, ys) = loop(xs)
+          val (siblings, zs) = loop(ys)
+          (MTree(x, children) :: siblings, zs)
+
+    loop(s)._1.head
+
+  def tree2String(t: MTree[Char]): String =
+    def loop(acc: DList[Char], t: MTree[Char]): DList[Char] =
+      val xs = t.children.foldLeft(DList.empty[Char])(loop)
+      val ys = DList.singleton(t.value) ++ xs ++ DList.singleton('^')
+      acc ++ ys
+
+    loop(DList.empty[Char], t).toList.mkString

mtree/src/P70C.scala

@@ -0,0 +1,10 @@
+package mtree
+
+// P70C (*) Count the nodes of a multiway tree.
+// Write a method nodeCount which counts the nodes of a given multiway tree.
+//
+// scala> MTree('a', List(MTree('f'))).nodeCount
+// res0: Int = 2
+object P70C:
+  extension [A](t: MTree[A])
+    def nodeCount: Int = 1 + t.children.foldLeft(0)((acc, c) => acc + c.nodeCount)

mtree/src/P71.scala

@@ -0,0 +1,25 @@
+package mtree
+
+// P71 (*) Determine the internal path length of a tree.
+// We define the internal path length of a multiway tree as the
+// total sum of the path lengths from the root to all nodes of the tree.
+// By this definition, the tree in the figure of problem P70 has an
+// internal path length of 9.  Write a method internalPathLength to return that sum.
+//
+// scala> "afg^^c^bd^e^^^".internalPathLength
+// res0: Int = 9
+/*
+ANSWER: We observe that the path length is equal to the number of
+nodes in a path from the root to a leaf, with a node counted only
+once. So, path length of abd = 3, but abd + abe is 4, not 6.
+
+The catch is to pass the _same_ accoumulated value to all the
+children of a node.
+ */
+object P71:
+  private def loop[A](acc: Int, t: MTree[A]): Int =
+    acc + t.children.map(loop(acc + 1, _)).sum
+
+  extension [A](t: MTree[A])
+    def internalPathLength: Int =
+      loop(0, t)

mtree/src/P72.scala

@@ -0,0 +1,16 @@
+package mtree
+
+// P72 (*) Construct the postorder sequence of the tree nodes.
+//
+// Write a method postorder which constructs the postorder sequence
+// of the nodes of a multiway tree.  The result should be a List.
+//
+// scala> "afg^^c^bd^e^^^".postorder
+// res0: List[Char] = List(g, f, c, d, e, b, a)
+object P72:
+  private def loop[A](acc: List[A], t: MTree[A]): List[A] =
+    t.children.foldRight(t.value :: acc)((tree, xs) => loop(xs, tree))
+
+  extension [A](t: MTree[A])
+    def postorder: List[A] =
+      loop(Nil, t)

mtree/src/P73.scala

@@ -0,0 +1,57 @@
+package mtree
+
+// P73 (**) Lisp-like tree representation.
+// There is a particular notation for multiway trees in Lisp.
+// Lisp is a prominent functional programming language.
+// In Lisp almost everything is a list.
+//
+// Our example tree would be represented in Lisp as (a (f g) c (b d e)).
+// The following pictures give some more examples.
+//
+// Note that in the "lispy" notation a node with successors (children) in
+// the tree is always the first element in a list, followed by its children.
+// The "lispy" representation of a multiway tree is a sequence of atoms and
+// parentheses '(' and ')', with the atoms separated by spaces.
+// We can represent this syntax as a Scala String.
+// Write a method lispyTree which constructs a "lispy string" from an MTree.
+//
+// scala> MTree("a", List(MTree("b", List(MTree("c"))))).lispyTree
+// res0: String = (a (b c))
+//
+// As a second, even more interesting, exercise try to write a method that
+// takes a "lispy" string and turns it into a multiway tree.
+object P73:
+  private def loop(s: String, i: Int): (MTree[Char], Int) =
+    if s(i).isSpaceChar then loop(s, i + 1)
+    else if s(i).isLetter then (MTree(s(i)), i + 1)
+    else if s(i) == '(' then
+      val x = s(i + 1)
+      val xs = IndexedSeq.unfold(i + 2)(j =>
+        Option.when(j < s.length() && s(j) != ')') {
+          val x = loop(s, j)
+          ((x, x._2))
+        }
+      )
+      (MTree(x, xs.map(_._1)), xs.last._2 + 1)
+    else throw IllegalStateException(s"s=$s, s($i)=${s(i)}")
+
+  def lispyString2Tree(s: String): MTree[Char] =
+    loop(s, 0)._1
+
+  extension [A](t: MTree[Char])
+    def lispyTree: String =
+      def loop(acc: DList[Char], t: MTree[Char]): DList[Char] =
+        if t.children.isEmpty then
+          acc
+            ++ DList.singleton(' ')
+            ++ DList.singleton(t.value)
+        else
+          val xs = t.children.foldLeft(DList.empty[Char])(loop)
+          acc
+            ++ DList.singleton(' ')
+            ++ DList.singleton('(')
+            ++ DList.singleton(t.value)
+            ++ xs
+            ++ DList.singleton(')')
+
+      loop(DList.empty[Char], t).toList.tail.mkString

mtree/src/package.scala

@@ -0,0 +1,3 @@
+package mtree
+
+given Conversion[String, MTree[Char]] = s => P70.string2MTree(s)

mtree/test/src/P70CSpec.scala

@@ -0,0 +1,9 @@
+package mtree
+
+import org.scalatest.funspec.AnyFunSpec
+import org.scalatest.matchers.should.Matchers.shouldBe
+import P70C.nodeCount
+
+class P70CSpec extends AnyFunSpec:
+  it("count the nodes of a multiway tree"):
+    MTree('a', List(MTree('f'))).nodeCount shouldBe 2