Scalaz（24）－ 泛函数据结构： Tree-数据游览及维护
上节我们讨论了Zipper－串形不可变集合（immutable sequencial collection）游标，在串形集合中左右游走及元素维护操作。这篇我们谈谈Tree。在电子商务应用中对于xml,json等格式文件的处理要求非常之普遍，scalaz提供了Tree数据类型及相关的游览及操作函数能更方便高效的处理xml,json文件及系统目录这些树形结构数据的相关编程。scalaz Tree的定义非常简单：scalaz/Tree.scala
* A multi-way tree, also known as a rose tree. Also known as Cofree[Stream, A]. */ sealed abstract class Tree[A] { import Tree._ /** The label at the root of this tree. */ def rootLabel: A /** The child nodes of this tree. */ def subForest: Stream[Tree[A]] ... Tree是由一个A值rootLabel及一个流中子树Stream[Tree[A]]组成。Tree可以只由一个A类型值rootLabel组成，这时流中子树subForest就是空的Stream.empty。只有rootLabel的Tree俗称叶（leaf），有subForest的称为节（node）。scalaz为任何类型提供了leaf和node的构建注入方法：syntax/TreeOps.scala
final class TreeOps[A](self: A) { def node(subForest: Tree[A]*): Tree[A] = Tree.node(self, subForest.toStream) def leaf: Tree[A] = Tree.leaf(self) } trait ToTreeOps { implicit def ToTreeOps[A](a: A) = new TreeOps(a) } 实际上注入方法调用了Tree里的构建函数：
trait TreeFunctions { /** Construct a new Tree node. */ def node[A](root: => A, forest: => Stream[Tree[A]]): Tree[A] = new Tree[A] { lazy val rootLabel = root lazy val subForest = forest override def toString = "<tree>" } /** Construct a tree node with no children. */ def leaf[A](root: => A): Tree[A] = node(root, Stream.empty) Tree提供了构建和模式拆分函数：
object Tree extends TreeInstances with TreeFunctions { /** Construct a tree node with no children. */ def apply[A](root: => A): Tree[A] = leaf(root) object Node { def unapply[A](t: Tree[A]): Option[(A, Stream[Tree[A]])] = Some((t.rootLabel, t.subForest)) } } 我们可以直接构建Tree：
1 Tree("ALeaf") === "ALeaf".leaf //> res5: Boolean = true 2 val tree: Tree[Int] = 3 1.node( 4 11.leaf, 5 12.node( 6 121.leaf), 7 2.node( 8 21.leaf, 9 22.leaf) 10 ) //> tree : scalaz.Tree[Int] = <tree> 11 tree.drawTree //> res6: String = "1 12 //| | 13 //| +- 11 14 //| | 15 //| +- 12 16 //| | | 17 //| | `- 121 18 //| | 19 //| `- 2 20 //| | 21 //| +- 21 22 //| | 23 //| `- 22 24 //| " Tree实现了下面众多的接口函数：
sealed abstract class TreeInstances { implicit val treeInstance: Traverse1[Tree] with Monad[Tree] with Comonad[Tree] with Align[Tree] with Zip[Tree] = new Traverse1[Tree] with Monad[Tree] with Comonad[Tree] with Align[Tree] with Zip[Tree] { def point[A](a: => A): Tree[A] = Tree.leaf(a) def cobind[A, B](fa: Tree[A])(f: Tree[A] => B): Tree[B] = fa cobind f def copoint[A](p: Tree[A]): A = p.rootLabel override def map[A, B](fa: Tree[A])(f: A => B) = fa map f def bind[A, B](fa: Tree[A])(f: A => Tree[B]): Tree[B] = fa flatMap f def traverse1Impl[G[_]: Apply, A, B](fa: Tree[A])(f: A => G[B]): G[Tree[B]] = fa traverse1 f override def foldRight[A, B](fa: Tree[A], z: => B)(f: (A, => B) => B): B = fa.foldRight(z)(f) override def foldMapRight1[A, B](fa: Tree[A])(z: A => B)(f: (A, => B) => B) = (fa.flatten.reverse: @unchecked) match { case h #:: t => t.foldLeft(z(h))((b, a) => f(a, b)) } override def foldLeft[A, B](fa: Tree[A], z: B)(f: (B, A) => B): B = fa.flatten.foldLeft(z)(f) override def foldMapLeft1[A, B](fa: Tree[A])(z: A => B)(f: (B, A) => B): B = fa.flatten match { case h #:: t => t.foldLeft(z(h))(f) } override def foldMap[A, B](fa: Tree[A])(f: A => B)(implicit F: Monoid[B]): B = fa foldMap f def alignWith[A, B, C](f: (/&/[A, B]) ⇒ C) = { def align(ta: Tree[A], tb: Tree[B]): Tree[C] = Tree.node(f(/&/(ta.rootLabel, tb.rootLabel)), Align[Stream].alignWith[Tree[A], Tree[B], Tree[C]]({ case /&/.This(sta) ⇒ sta map {a ⇒ f(/&/.This(a))} case /&/.That(stb) ⇒ stb map {b ⇒ f(/&/.That(b))} case /&/.Both(sta, stb) ⇒ align(sta, stb) })(ta.subForest, tb.subForest)) align _ } def zip[A, B](aa: => Tree[A], bb: => Tree[B]) = { val a = aa val b = bb Tree.node( (a.rootLabel, b.rootLabel), Zip[Stream].zipWith(a.subForest, b.subForest)(zip(_, _)) ) } } implicit def treeEqual[A](implicit A0: Equal[A]): Equal[Tree[A]] = new TreeEqual[A] { def A = A0 } implicit def treeOrder[A](implicit A0: Order[A]): Order[Tree[A]] = new Order[Tree[A]] with TreeEqual[A] { def A = A0 import std.stream._ override def order(x: Tree[A], y: Tree[A]) = A.order(x.rootLabel, y.rootLabel) match { case Ordering.EQ => Order[Stream[Tree[A]]].order(x.subForest, y.subForest) case x => x } } 那么Tree就是个Monad，也是Functor,Applicative，还是traversable,foldable。Tree也实现了Order,Equal实例，可以进行值的顺序比较。我们就用些例子来说明吧：
1 // 是 Functor... 2 (tree map { v: Int => v + 1 }) === 3 2.node( 4 12.leaf, 5 13.node( 6 122.leaf), 7 3.node( 8 22.leaf, 9 23.leaf) 10 ) //> res7: Boolean = true 11 12 // ...是 Monad 13 1.point[Tree] === 1.leaf //> res8: Boolean = true 14 val t2 = tree >>= (x => (x == 2) ? x.leaf | x.node((-x).leaf)) 15 //> t2 : scalaz.Tree[Int] = <tree> 16 t2 === 1.node((-1).leaf, 2.leaf, 3.node((-3).leaf, 4.node((-4).leaf))) 17 //> res9: Boolean = false 18 t2.drawTree //> res10: String = "1 19 //| | 20 //| +- -1 21 //| | 22 //| +- 11 23 //| | | 24 //| | `- -11 25 //| | 26 //| +- 12 27 //| | | 28 //| | +- -12 29 //| | | 30 //| | `- 121 31 //| | | 32 //| | `- -121 33 //| | 34 //| `- 2 35 //| | 36 //| +- 21 37 //| | | 38 //| | `- -21 39 //| | 40 //| `- 22 41 //| | 42 //| `- -22 43 //| " 44 // ...是 Foldable 45 tree.foldMap(_.toString) === "1111212122122" //> res11: Boolean = true 说到构建Tree，偶然在网上发现了这么一个Tree构建函数：
def pathTree[E](root: E, paths: Seq[Seq[E]]): Tree[E] = { root.node(paths groupBy (_.head) map { case (parent, subpaths) => pathTree(parent, subpaths collect { case pp +: rest if rest.nonEmpty => rest }) } toSeq: _*) } 据说这个pathTree函数能把List里的目录结构转化成Tree。先看看到底是不是具备如此功能：
1 val paths = List(List("A","a1","a2"),List("B","b1")) 2 //> paths : List[List[String]] = List(List(A, a1, a2), List(B, b1)) 3 pathTree("root",paths) drawTree //> res0: String = ""root" 4 //| | 5 //| +- "A" 6 //| | | 7 //| | `- "a1" 8 //| | | 9 //| | `- "a2" 10 //| | 11 //| `- "B" 12 //| | 13 //| `- "b1" 14 //| " 15 val paths = List(List("A","a1","a2"),List("B","b1"),List("B","b2","b3")) 16 //> paths : List[List[String]] = List(List(A, a1, a2), List(B, b1), List(B, b2, 17 //| b3)) 18 pathTree("root",paths) drawTree //> res0: String = ""root" 19 //| | 20 //| +- "A" 21 //| | | 22 //| | `- "a1" 23 //| | | 24 //| | `- "a2" 25 //| | 26 //| `- "B" 27 //| | 28 //| +- "b2" 29 //| | | 30 //| | `- "b3" 31 //| | 32 //| `- "b1" 33 //| " 果然能行，而且还能把"B"节点合并汇集。这个函数的作者简直就是个神人，起码是个算法和FP语法运用大师。我虽然还无法达到大师的程度能写出这样的泛函程序，但好奇心是挡不住的，总想了解这个函数是怎么运作的。可以用一些测试数据来逐步跟踪一下：
1 val paths = List(List("A")) //> paths : List[List[String]] = List(List(A)) 2 val gpPaths =paths.groupBy(_.head) //> gpPaths : scala.collection.immutable.Map[String,List[List[String]]] = Map(A-> List(List(A))) 3 List(List("A")) collect { case pp +: rest if rest.nonEmpty => rest } 4 //> res0: List[List[String]] = List() 通过上面的跟踪约化我们看到List(List(A))在pathTree里的执行过程。这里把复杂的groupBy和collect函数的用法和结果了解了。实际上整个过程相当于：
1 "root".node( 2 "A".node(List().toSeq: _*) 3 ) drawTree //> res3: String = ""root" 4 //| | 5 //| `- "A" 6 //| "
如果再增加一个点就相当于：
1 "root".node( 2 "A".node(List().toSeq: _*), 3 "B".node(List().toSeq: _*) 4 ) drawTree //> res4: String = ""root" 5 //| | 6 //| +- "A" 7 //| | 8 //| `- "B" 9 //| "
加多一层：
1 val paths = List(List("A","a1")) //> paths : List[List[String]] = List(List(A, a1)) 2 val gpPaths =paths.groupBy(_.head) //> gpPaths : scala.collection.immutable.Map[String,List[List[String]]] = Map(A 3 //| -> List(List(A, a1))) 4 List(List("A","a1")) collect { case pp +: rest if rest.nonEmpty => rest } 5 //> res0: List[List[String]] = List(List(a1)) 6 7 //化解成 8 "root".node( 9 "A".node( 10 "a1".node( 11 List().toSeq: _*) 12 ) 13 ) drawTree //> res3: String = ""root" 14 //| | 15 //| `- "A" 16 //| | 17 //| `- "a1" 18 //| " 合并目录：
1 val paths = List(List("A","a1"),List("A","a2")) //> paths : List[List[String]] = List(List(A, a1), List(A, a2)) 2 val gpPaths =paths.groupBy(_.head) //> gpPaths : scala.collection.immutable.Map[String,List[List[String]]] = Map(A 3 //| -> List(List(A, a1), List(A, a2))) 4 List(List("A","a1"),List("A","a2")) collect { case pp +: rest if rest.nonEmpty => rest } 5 //> res0: List[List[String]] = List(List(a1), List(a2)) 6 7 //相当产生结果 8 "root".node( 9 "A".node( 10 "a1".node( 11 List().toSeq: _*) 12 , 13 "a2".node( 14 List().toSeq: _*) 15 ) 16 ) drawTree //> res3: String = ""root" 17 //| | 18 //| `- "A" 19 //| | 20 //| +- "a1" 21 //| | 22 //| `- "a2" 23 //| " 相信这些跟踪过程足够了解整个函数的工作原理了。
有了Tree构建方法后就需要Tree的游动和操作函数了。与串形集合的直线游动不同的是，树形集合游动方式是分岔的。所以Zipper不太适用于树形结构。scalaz特别提供了树形集合的定位游标TreeLoc，我们看看它的定义：scalaz/TreeLoc.scala
final case class TreeLoc[A](tree: Tree[A], lefts: TreeForest[A], rights: TreeForest[A], parents: Parents[A]) { ... trait TreeLocFunctions { type TreeForest[A] = Stream[Tree[A]] type Parent[A] = (TreeForest[A], A, TreeForest[A]) type Parents[A] = Stream[Parent[A]] 树形集合游标TreeLoc由当前节点tree、左子树lefts、右子树rights及父树parents组成。lefts,rights,parents都是在流中的树形Stream[Tree[A]]。
用Tree.loc可以直接对目标树生成TreeLoc：
1 /** A TreeLoc zipper of this tree, focused on the root node. */ 2 def loc: TreeLoc[A] = TreeLoc.loc(this, Stream.Empty, Stream.Empty, Stream.Empty) 3 4 val tree: Tree[Int] = 5 1.node( 6 11.leaf, 7 12.node( 8 121.leaf), 9 2.node( 10 21.leaf, 11 22.leaf) 12 ) //> tree : scalaz.Tree[Int] = <tree> 13 14 tree.loc //> res7: scalaz.TreeLoc[Int] = TreeLoc(<tree>,Stream(),Stream(),Stream())
TreeLoc的游动函数：
def root: TreeLoc[A] = parent match { case Some(z) => z.root case None => this } /** Select the left sibling of the current node. */ def left: Option[TreeLoc[A]] = lefts match { case t #:: ts => Some(loc(t, ts, tree #:: rights, parents)) case Stream.Empty => None } /** Select the right sibling of the current node. */ def right: Option[TreeLoc[A]] = rights match { case t #:: ts => Some(loc(t, tree #:: lefts, ts, parents)) case Stream.Empty => None } /** Select the leftmost child of the current node. */ def firstChild: Option[TreeLoc[A]] = tree.subForest match { case t #:: ts => Some(loc(t, Stream.Empty, ts, downParents)) case Stream.Empty => None } /** Select the rightmost child of the current node. */ def lastChild: Option[TreeLoc[A]] = tree.subForest.reverse match { case t #:: ts => Some(loc(t, ts, Stream.Empty, downParents)) case Stream.Empty => None } /** Select the nth child of the current node. */ def getChild(n: Int): Option[TreeLoc[A]] = for {lr <- splitChildren(Stream.Empty, tree.subForest, n) ls = lr._1 } yield loc(ls.head, ls.tail, lr._2, downParents) 我们试着用这些函数游动：
1 val tree: Tree[Int] = 2 1.node( 3 11.leaf, 4 12.node( 5 121.leaf), 6 2.node( 7 21.leaf, 8 22.leaf) 9 ) //> tree : scalaz.Tree[Int] = <tree> 10 tree.loc //> res7: scalaz.TreeLoc[Int] = TreeLoc(<tree>,Stream(),Stream(),Stream()) 11 val l = for { 12 l1 <- tree.loc.some 13 l2 <- l1.firstChild 14 l3 <- l1.lastChild 15 l4 <- l3.firstChild 16 } yield (l1,l2,l3,l4) //> l : Option[(scalaz.TreeLoc[Int], scalaz.TreeLoc[Int], scalaz.TreeLoc[Int], 17 //| scalaz.TreeLoc[Int])] = Some((TreeLoc(<tree>,Stream(),Stream(),Stream()),T 18 //| reeLoc(<tree>,Stream(),Stream(<tree>, <tree>),Stream((Stream(),1,Stream()), 19 //| ?)),TreeLoc(<tree>,Stream(<tree>, <tree>),Stream(),Stream((Stream(),1,Stre 20 //| am()), ?)),TreeLoc(<tree>,Stream(),Stream(<tree>, ?),Stream((Stream(<tree>, 21 //| <tree>),2,Stream()), ?)))) 22 23 l.get._1.getLabel //> res8: Int = 1 24 l.get._2.getLabel //> res9: Int = 11 25 l.get._3.getLabel //> res10: Int = 2 26 l.get._4.getLabel //> res11: Int = 21 跳动函数：
/** Select the nth child of the current node. */ def getChild(n: Int): Option[TreeLoc[A]] = for {lr <- splitChildren(Stream.Empty, tree.subForest, n) ls = lr._1 } yield loc(ls.head, ls.tail, lr._2, downParents) /** Select the first immediate child of the current node that satisfies the given predicate. */ def findChild(p: Tree[A] => Boolean): Option[TreeLoc[A]] = { @tailrec def split(acc: TreeForest[A], xs: TreeForest[A]): Option[(TreeForest[A], Tree[A], TreeForest[A])] = (acc, xs) match { case (acc, Stream.cons(x, xs)) => if (p(x)) Some((acc, x, xs)) else split(Stream.cons(x, acc), xs) case _ => None } for (ltr <- split(Stream.Empty, tree.subForest)) yield loc(ltr._2, ltr._1, ltr._3, downParents) } /**Select the first descendant node of the current node that satisfies the given predicate. */ def find(p: TreeLoc[A] => Boolean): Option[TreeLoc[A]] = Cobind[TreeLoc].cojoin(this).tree.flatten.find(p) find用法示范：
1 val tree: Tree[Int] = 2 1.node( 3 11.leaf, 4 12.node( 5 121.leaf), 6 2.node( 7 21.leaf, 8 22.leaf) 9 ) //> tree : scalaz.Tree[Int] = <tree> 10 tree.loc //> res7: scalaz.TreeLoc[Int] = TreeLoc(<tree>,Stream(),Stream(),Stream()) 11 val l = for { 12 l1 <- tree.loc.some 13 l2 <- l1.find{_.getLabel == 2} 14 l3 <- l1.find{_.getLabel == 121} 15 l4 <- l2.find{_.getLabel == 22} 16 l5 <- l1.findChild{_.rootLabel == 12} 17 l6 <- l1.findChild{_.rootLabel == 2} 18 } yield l6 //> l : Option[scalaz.TreeLoc[Int]] = Some(TreeLoc(<tree>,Stream(<tree>, ?),St 19 //| ream(),Stream((Stream(),1,Stream()), ?))) 注意：上面6个跳动都成功了。如果无法跳转结果会是None
insert,modify,delete这些操作函数：
/** Replace the current node with the given one. */ def setTree(t: Tree[A]): TreeLoc[A] = loc(t, lefts, rights, parents) /** Modify the current node with the given function. */ def modifyTree(f: Tree[A] => Tree[A]): TreeLoc[A] = setTree(f(tree)) /** Modify the label at the current node with the given function. */ def modifyLabel(f: A => A): TreeLoc[A] = setLabel(f(getLabel)) /** Get the label of the current node. */ def getLabel: A = tree.rootLabel /** Set the label of the current node. */ def setLabel(a: A): TreeLoc[A] = modifyTree((t: Tree[A]) => node(a, t.subForest)) /** Insert the given node to the left of the current node and give it focus. */ def insertLeft(t: Tree[A]): TreeLoc[A] = loc(t, lefts, Stream.cons(tree, rights), parents) /** Insert the given node to the right of the current node and give it focus. */ def insertRight(t: Tree[A]): TreeLoc[A] = loc(t, Stream.cons(tree, lefts), rights, parents) /** Insert the given node as the first child of the current node and give it focus. */ def insertDownFirst(t: Tree[A]): TreeLoc[A] = loc(t, Stream.Empty, tree.subForest, downParents) /** Insert the given node as the last child of the current node and give it focus. */ def insertDownLast(t: Tree[A]): TreeLoc[A] = loc(t, tree.subForest.reverse, Stream.Empty, downParents) /** Insert the given node as the nth child of the current node and give it focus. */ def insertDownAt(n: Int, t: Tree[A]): Option[TreeLoc[A]] = for (lr <- splitChildren(Stream.Empty, tree.subForest, n)) yield loc(t, lr._1, lr._2, downParents) /** Delete the current node and all its children. */ def delete: Option[TreeLoc[A]] = rights match { case Stream.cons(t, ts) => Some(loc(t, lefts, ts, parents)) case _ => lefts match { case Stream.cons(t, ts) => Some(loc(t, ts, rights, parents)) case _ => for (loc1 <- parent) yield loc1.modifyTree((t: Tree[A]) => node(t.rootLabel, Stream.Empty)) } } 用法示范：
1 val tr = 1.leaf //> tr : scalaz.Tree[Int] = <tree> 2 val tl = for { 3 l1 <- tr.loc.some 4 l3 <- l1.insertDownLast(12.leaf).some 5 l4 <- l3.insertDownLast(121.leaf).some 6 l5 <- l4.root.some 7 l2 <- l5.insertDownFirst(11.leaf).some 8 l6 <- l2.root.some 9 l7 <- l6.find{_.getLabel == 12} 10 l8 <- l7.setLabel(102).some 11 } yield l8 //> tl : Option[scalaz.TreeLoc[Int]] = Some(TreeLoc(<tree>,Stream(<tree>, ?),S 12 //| tream(),Stream((Stream(),1,Stream()), ?))) 13 14 tl.get.toTree.drawTree //> res8: String = "1 15 //| | 16 //| +- 11 17 //| | 18 //| `- 102 19 //| | 20 //| `- 121 21 //| " 22 setTree和delete会替换当前节点下的所有子树：
1 val tree: Tree[Int] = 2 1.node( 3 11.leaf, 4 12.node( 5 121.leaf), 6 2.node( 7 21.leaf, 8 22.leaf) 9 ) //> tree : scalaz.Tree[Int] = <tree> 10 def modTree(t: Tree[Int]): Tree[Int] = { 11 val l = for { 12 l1 <- t.loc.some 13 l2 <- l1.find{_.getLabel == 22} 14 l3 <- l2.setTree { 3.node (31.leaf) }.some 15 } yield l3 16 l.get.toTree 17 } //> modTree: (t: scalaz.Tree[Int])scalaz.Tree[Int] 18 val l = for { 19 l1 <- tree.loc.some 20 l2 <- l1.find{_.getLabel == 2} 21 l3 <- l2.modifyTree{modTree(_)}.some 22 l4 <- l3.root.some 23 l5 <- l4.find{_.getLabel == 12} 24 l6 <- l5.delete 25 } yield l6 //> l : Option[scalaz.TreeLoc[Int]] = Some(TreeLoc(<tree>,Stream(<tree>, ?),St 26 //| ream(),Stream((Stream(),1,Stream()), ?))) 27 l.get.toTree.drawTree //> res7: String = "1 28 //| | 29 //| +- 11 30 //| | 31 //| `- 2 32 //| | 33 //| +- 21 34 //| | 35 //| `- 3 36 //| | 37 //| `- 31 38 //| " 通过scalaz的Tree和TreeLoc数据结构，以及一整套树形结构游览、操作函数，我们可以方便有效地实现FP风格的不可变树形集合编程。
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