inorder + preorder如何构造唯一的二叉树？

Question

最近，我的问题被标记为重复，例如this，即使它们不是。所以，让我先从下面开始，然后我将解释我的问题。

为什么这个问题不重复？

我没有询问如何在顺序和创建时创建二叉树给出了前序遍历。我要求证明，inorder + preorder遍历定义了一个唯一的二叉树。

现在，原始问题。我去面试，面试官问我这个问题。我被困住了，无法继续。 ：|

问题：鉴于此问题＆amp; preorder遍历二叉树。证明给定数据只有一个二叉树。换句话说，证明两个不同的二元树不能具有相同的顺序和优点。预先遍历。假设树中的所有元素都是唯一的（感谢@envy_intelligence指出这个假设）。

我尝试使用示例说服采访者，但采访者要求数学/直觉证明。任何人都可以帮我证明吗？

Answer 1

从前序遍历开始。它是空的，在这种情况下你已经完成，或者它有第一个元素r0，树的根。现在搜索r0的inorder遍历。左子树将全部出现在该点之前，右子树将全部出现在该点之后。因此，您可以将该点的inorder遍历划分为左子树il的inorder遍历和右子树ir的inorder遍历。

如果il为空，则前序遍历的其余部分属于右子树，您可以继续归纳。如果ir为空，则另一方面也会发生同样的事情。如果两者都不为空，则在前序遍历的其余部分中找到ir的第一个元素。这将它划分为左子树的前序遍历和右子树中的一个。立即归纳。

如果有人对正式证明感兴趣，我（最终）设法在Idris中制作了一个。但是，我还没有花时间尝试使其非常易读，因此实际上很难阅读它。我建议您主要查看顶级类型（即，引理，定理和定义），并尽量避免在证明（术语）中陷入困境。

首先是一些预赛：

module PreIn
import Data.List
%default total

现在第一个真正的想法：二叉树。

data Tree : Type -> Type where
  Tip : Tree a
  Node : (l : Tree a) -> (v : a) -> (r : Tree a) -> Tree a
%name Tree t, u

现在是第二个重要的想法：在特定树中找到特定元素的方法的想法。这与Elem中的Data.List类型密切相关，后者表达了查找特定列表中特定元素的方法。

data InTree : a -> Tree a -> Type where
  AtRoot : x `InTree` (Node l x r)
  OnLeft : x `InTree` l -> x `InTree` (Node l v r)
  OnRight : x `InTree` r -> x `InTree` (Node l v r)

然后有一大堆可怕的引理，其中有几条由Eric Mertens（glguy）在his answer中提出我的问题。

可怕的外表

size : Tree a -> Nat
size Tip = Z
size (Node l v r) = size l + (S Z + size r)

onLeftInjective : OnLeft p = OnLeft q -> p = q
onLeftInjective Refl = Refl

onRightInjective : OnRight p = OnRight q -> p = q
onRightInjective Refl = Refl

inorder : Tree a -> List a
inorder Tip = []
inorder (Node l v r) = inorder l ++ [v] ++ inorder r

instance Uninhabited (Here = There y) where
  uninhabited Refl impossible

instance Uninhabited (x `InTree` Tip) where
  uninhabited AtRoot impossible

elemAppend : {x : a} -> (ys,xs : List a) -> x `Elem` xs -> x `Elem` (ys ++ xs)
elemAppend [] xs xInxs = xInxs
elemAppend (y :: ys) xs xInxs = There (elemAppend ys xs xInxs)

appendElem : {x : a} -> (xs,ys : List a) -> x `Elem` xs -> x `Elem` (xs ++ ys)
appendElem (x :: zs) ys Here = Here
appendElem (y :: zs) ys (There pr) = There (appendElem zs ys pr)

tThenInorder : {x : a} -> (t : Tree a) -> x `InTree` t -> x `Elem` inorder t
tThenInorder (Node l x r) AtRoot = elemAppend _ _ Here
tThenInorder (Node l v r) (OnLeft pr) = appendElem _ _ (tThenInorder _ pr)
tThenInorder (Node l v r) (OnRight pr) = elemAppend _ _ (There (tThenInorder _ pr))

listSplit_lem : (x,z : a) -> (xs,ys:List a) -> Either (x `Elem` xs) (x `Elem` ys)
  -> Either (x `Elem` (z :: xs)) (x `Elem` ys)
listSplit_lem x z xs ys (Left prf) = Left (There prf)
listSplit_lem x z xs ys (Right prf) = Right prf


listSplit : {x : a} -> (xs,ys : List a) -> x `Elem` (xs ++ ys) -> Either (x `Elem` xs) (x `Elem` ys)
listSplit [] ys xelem = Right xelem
listSplit (z :: xs) ys Here = Left Here
listSplit {x} (z :: xs) ys (There pr) = listSplit_lem x z xs ys (listSplit xs ys pr)

mutual
  inorderThenT : {x : a} -> (t : Tree a) -> x `Elem` inorder t -> InTree x t
  inorderThenT Tip xInL = absurd xInL
  inorderThenT {x} (Node l v r) xInL = inorderThenT_lem x l v r xInL (listSplit (inorder l) (v :: inorder r) xInL)

  inorderThenT_lem : (x : a) ->
                     (l : Tree a) -> (v : a) -> (r : Tree a) ->
                     x `Elem` inorder (Node l v r) ->
                     Either (x `Elem` inorder l) (x `Elem` (v :: inorder r)) ->
                     InTree x (Node l v r)
  inorderThenT_lem x l v r xInL (Left locl) = OnLeft (inorderThenT l locl)
  inorderThenT_lem x l x r xInL (Right Here) = AtRoot
  inorderThenT_lem x l v r xInL (Right (There locr)) = OnRight (inorderThenT r locr)

unsplitRight : {x : a} -> (e : x `Elem` ys) -> listSplit xs ys (elemAppend xs ys e) = Right e
unsplitRight {xs = []} e = Refl
unsplitRight {xs = (x :: xs)} e = rewrite unsplitRight {xs} e in Refl

unsplitLeft : {x : a} -> (e : x `Elem` xs) -> listSplit xs ys (appendElem xs ys e) = Left e
unsplitLeft {xs = []} Here impossible
unsplitLeft {xs = (x :: xs)} Here = Refl
unsplitLeft {xs = (x :: xs)} {ys} (There pr) =
  rewrite unsplitLeft {xs} {ys} pr in Refl

splitLeft_lem1 : (Left (There w) = listSplit_lem x y xs ys (listSplit xs ys z)) ->
                 (Left w = listSplit xs ys z) 

splitLeft_lem1 {w} {xs} {ys} {z} prf with (listSplit xs ys z)
  splitLeft_lem1 {w}  Refl | (Left w) = Refl
  splitLeft_lem1 {w}  Refl | (Right s) impossible

splitLeft_lem2 : Left Here = listSplit_lem x x xs ys (listSplit xs ys z) -> Void
splitLeft_lem2 {x} {xs} {ys} {z} prf with (listSplit xs ys z)
  splitLeft_lem2 {x = x} {xs = xs} {ys = ys} {z = z} Refl | (Left y) impossible
  splitLeft_lem2 {x = x} {xs = xs} {ys = ys} {z = z} Refl | (Right y) impossible

splitLeft : {x : a} -> (xs,ys : List a) ->
            (loc : x `Elem` (xs ++ ys)) ->
            Left e = listSplit {x} xs ys loc ->
            appendElem {x} xs ys e = loc
splitLeft {e} [] ys loc prf = absurd e
splitLeft (x :: xs) ys Here prf = rewrite leftInjective prf in Refl
splitLeft {e = Here} (x :: xs) ys (There z) prf = absurd (splitLeft_lem2 prf)
splitLeft {e = (There w)} (y :: xs) ys (There z) prf =
  cong $ splitLeft xs ys z (splitLeft_lem1 prf)

splitMiddle_lem3 : Right Here = listSplit_lem y x xs (y :: ys) (listSplit xs (y :: ys) z) ->
                   Right Here = listSplit xs (y :: ys) z

splitMiddle_lem3 {y} {x} {xs} {ys} {z} prf with (listSplit xs (y :: ys) z)
  splitMiddle_lem3 {y = y} {x = x} {xs = xs} {ys = ys} {z = z} Refl | (Left w) impossible
  splitMiddle_lem3 {y = y} {x = x} {xs = xs} {ys = ys} {z = z} prf | (Right w) =
    cong $ rightInjective prf  -- This funny dance strips the Rights off and then puts them
                               -- back on so as to change type.


splitMiddle_lem2 : Right Here = listSplit xs (y :: ys) pl ->
                   elemAppend xs (y :: ys) Here = pl

splitMiddle_lem2 {xs} {y} {ys} {pl} prf with (listSplit xs (y :: ys) pl) proof prpr
  splitMiddle_lem2 {xs = xs} {y = y} {ys = ys} {pl = pl} Refl | (Left loc) impossible
  splitMiddle_lem2 {xs = []} {y = y} {ys = ys} {pl = pl} Refl | (Right Here) = rightInjective prpr
  splitMiddle_lem2 {xs = (x :: xs)} {y = x} {ys = ys} {pl = Here} prf | (Right Here) = (\Refl impossible) prpr
  splitMiddle_lem2 {xs = (x :: xs)} {y = y} {ys = ys} {pl = (There z)} prf | (Right Here) =
    cong $ splitMiddle_lem2 {xs} {y} {ys} {pl = z} (splitMiddle_lem3 prpr)

splitMiddle_lem1 : Right Here = listSplit_lem y x xs (y :: ys) (listSplit xs (y :: ys) pl) ->
                   elemAppend xs (y :: ys) Here = pl

splitMiddle_lem1 {y} {x} {xs} {ys} {pl} prf with (listSplit xs (y :: ys) pl) proof prpr
  splitMiddle_lem1 {y = y} {x = x} {xs = xs} {ys = ys} {pl = pl} Refl | (Left z) impossible
  splitMiddle_lem1 {y = y} {x = x} {xs = xs} {ys = ys} {pl = pl} Refl | (Right Here) = splitMiddle_lem2 prpr

splitMiddle : Right Here = listSplit xs (y::ys) loc ->
              elemAppend xs (y::ys) Here = loc

splitMiddle {xs = []} prf = rightInjective prf
splitMiddle {xs = (x :: xs)} {loc = Here} Refl impossible
splitMiddle {xs = (x :: xs)} {loc = (There y)} prf = cong $ splitMiddle_lem1 prf

splitRight_lem1 : Right (There pl) = listSplit (q :: xs) (y :: ys) (There z) ->
                  Right (There pl) = listSplit xs (y :: ys) z

splitRight_lem1 {xs} {ys} {y} {z} prf with (listSplit xs (y :: ys) z)
  splitRight_lem1 {xs = xs} {ys = ys} {y = y} {z = z} Refl | (Left x) impossible
  splitRight_lem1 {xs = xs} {ys = ys} {y = y} {z = z} prf | (Right x) =
    cong $ rightInjective prf  -- Type dance: take the Right off and put it back on.

splitRight : Right (There pl) = listSplit xs (y :: ys) loc ->
             elemAppend xs (y :: ys) (There pl) = loc
splitRight {pl = pl} {xs = []} {y = y} {ys = ys} {loc = loc} prf = rightInjective prf
splitRight {pl = pl} {xs = (x :: xs)} {y = y} {ys = ys} {loc = Here} Refl impossible
splitRight {pl = pl} {xs = (x :: xs)} {y = y} {ys = ys} {loc = (There z)} prf =
  let rec = splitRight {pl} {xs} {y} {ys} {loc = z} in cong $ rec (splitRight_lem1 prf)

树与其顺序遍历之间的对应关系

这些可怕的引理导致以下关于顺序遍历的定理，这些定理一起展示了在树中找到特定元素的方法与在其顺序遍历中找到该元素的方法之间的一对一对应关系。

---------------------------
-- tThenInorder is a bijection from ways to find a particular element in a tree
-- and ways to find that element in its inorder traversal. `inorderToFro`
-- and `inorderFroTo` together demonstrate this by showing that `inorderThenT` is
-- its inverse.

||| `tThenInorder t` is a retraction of `inorderThenT t`
inorderFroTo : {x : a} -> (t : Tree a) -> (loc : x `Elem` inorder t) -> tThenInorder t (inorderThenT t loc) = loc
inorderFroTo Tip loc = absurd loc
inorderFroTo (Node l v r) loc with (listSplit (inorder l) (v :: inorder r) loc) proof prf
  inorderFroTo (Node l v r) loc | (Left here) =
    rewrite inorderFroTo l here in splitLeft _ _ loc prf
  inorderFroTo (Node l v r) loc | (Right Here) = splitMiddle prf
  inorderFroTo (Node l v r) loc | (Right (There x)) =
    rewrite inorderFroTo r x in splitRight prf

||| `inorderThenT t` is a retraction of `tThenInorder t`
inorderToFro : {x : a} -> (t : Tree a) -> (loc : x `InTree` t) -> inorderThenT t (tThenInorder t loc) = loc
inorderToFro (Node l v r) (OnLeft xInL) =
  rewrite unsplitLeft {ys = v :: inorder r} (tThenInorder l xInL)
  in cong $ inorderToFro _ xInL
inorderToFro (Node l x r) AtRoot =
  rewrite unsplitRight {x} {xs = inorder l} {ys = x :: inorder r} (tThenInorder (Node Tip x r) AtRoot)
  in Refl
inorderToFro {x} (Node l v r) (OnRight xInR) =
  rewrite unsplitRight {x} {xs = inorder l} {ys = v :: inorder r} (tThenInorder (Node Tip v r) (OnRight xInR))
  in cong $ inorderToFro _ xInR

树与其前序遍历之间的对应关系

然后可以使用许多相同的引理来证明前序遍历的相应定理：

preorder : Tree a -> List a
preorder Tip = []
preorder (Node l v r) = v :: (preorder l ++ preorder r)

tThenPreorder : (t : Tree a) -> x `InTree` t -> x `Elem` preorder t
tThenPreorder Tip AtRoot impossible
tThenPreorder (Node l x r) AtRoot = Here
tThenPreorder (Node l v r) (OnLeft loc) = appendElem _ _ (There (tThenPreorder _ loc))
tThenPreorder (Node l v r) (OnRight loc) = elemAppend (v :: preorder l) (preorder r) (tThenPreorder _ loc)

mutual
  preorderThenT : (t : Tree a) -> x `Elem` preorder t -> x `InTree` t
  preorderThenT {x = x} (Node l x r) Here = AtRoot
  preorderThenT {x = x} (Node l v r) (There y) = preorderThenT_lem (listSplit _ _ y)

  preorderThenT_lem : Either (x `Elem` preorder l) (x `Elem` preorder r) -> x `InTree` (Node l v r)
  preorderThenT_lem {x = x} {l = l} {v = v} {r = r} (Left lloc) = OnLeft (preorderThenT l lloc)
  preorderThenT_lem {x = x} {l = l} {v = v} {r = r} (Right rloc) = OnRight (preorderThenT r rloc)

splitty : Right pl = listSplit xs ys loc -> elemAppend xs ys pl = loc
splitty {pl = Here} {xs = xs} {ys = (x :: zs)} {loc = loc} prf = splitMiddle prf
splitty {pl = (There x)} {xs = xs} {ys = (y :: zs)} {loc = loc} prf = splitRight prf

preorderFroTo : {x : a} -> (t : Tree a) -> (loc : x `Elem` preorder t) ->
                tThenPreorder t (preorderThenT t loc) = loc
preorderFroTo Tip Here impossible
preorderFroTo (Node l x r) Here = Refl
preorderFroTo (Node l v r) (There loc) with (listSplit (preorder l) (preorder r) loc) proof spl
  preorderFroTo (Node l v r) (There loc) | (Left pl) =
    rewrite sym (splitLeft {e=pl} (preorder l) (preorder r) loc spl)
    in cong {f = There} $ cong {f = appendElem (preorder l) (preorder r)} (preorderFroTo _ _)
  preorderFroTo (Node l v r) (There loc) | (Right pl) =
      rewrite preorderFroTo r pl in cong {f = There} (splitty spl)

preorderToFro : {x : a} -> (t : Tree a) -> (loc : x `InTree` t) -> preorderThenT t (tThenPreorder t loc) = loc
preorderToFro (Node l x r) AtRoot = Refl
preorderToFro (Node l v r) (OnLeft loc) =
  rewrite unsplitLeft {ys = preorder r} (tThenPreorder l loc)
  in cong {f = OnLeft} (preorderToFro l loc)
preorderToFro (Node l v r) (OnRight loc) =
  rewrite unsplitRight {xs = preorder l} (tThenPreorder r loc)
  in cong {f = OnRight} (preorderToFro r loc)

到目前为止好吗？很高兴听见。你寻求的定理快到了！首先，我们需要一个树是“内射”的概念，我认为这是在这种情况下“没有重复”的最简单的概念。如果你不喜欢这个概念，不要担心;南方还有一个方向。这个说t树只有loc1和loc1是x t中找到值loc1的方法，loc2必须是唯一的等于InjTree : Tree a -> Type InjTree t = (x : a) -> (loc1, loc2 : x `InTree` t) -> loc1 = loc2 。

InjList : List a -> Type
InjList xs = (x : a) -> (loc1, loc2 : x `Elem` xs) -> loc1 = loc2

||| If a tree is injective, so is its preorder traversal
treePreInj : (t : Tree a) -> InjTree t -> InjList (preorder t)
treePreInj {a} t it x loc1 loc2 =
  let foo = preorderThenT {a} {x} t loc1
      bar = preorderThenT {a} {x} t loc2
      baz = it x foo bar
  in rewrite sym $ preorderFroTo t loc1
  in rewrite sym $ preorderFroTo t loc2
  in cong baz

||| If a tree is injective, so is its inorder traversal
treeInInj : (t : Tree a) -> InjTree t -> InjList (inorder t)
treeInInj {a} t it x loc1 loc2 =
  let foo = inorderThenT {a} {x} t loc1
      bar = inorderThenT {a} {x} t loc2
      baz = it x foo bar
  in rewrite sym $ inorderFroTo t loc1
  in rewrite sym $ inorderFroTo t loc2
  in cong baz

||| If a tree's preorder traversal is injective, so is the tree.
injPreTree : (t : Tree a) -> InjList (preorder t) -> InjTree t
injPreTree {a} t il x loc1 loc2 =
  let
    foo = tThenPreorder {a} {x} t loc1
    bar = tThenPreorder {a} {x} t loc2
    baz = il x foo bar
  in rewrite sym $ preorderToFro t loc1
  in rewrite sym $ preorderToFro t loc2
  in cong baz

||| If a tree's inorder traversal is injective, so is the tree.
injInTree : (t : Tree a) -> InjList (inorder t) -> InjTree t
injInTree {a} t il x loc1 loc2 =
  let
    foo = tThenInorder {a} {x} t loc1
    bar = tThenInorder {a} {x} t loc2
    baz = il x foo bar
  in rewrite sym $ inorderToFro t loc1
  in rewrite sym $ inorderToFro t loc2
  in cong baz

我们也想要列表的相应概念，因为我们将证明树是唯一的，只有当它们的遍历是。这些证据就在下面，并按照前面的说法进行。

headsSame : {x:a} -> {xs : List a} -> {y : a} -> {ys : List a} -> (x :: xs) = (y :: ys) -> x = y
headsSame Refl = Refl

tailsSame : {x:a} -> {xs : List a} -> {y : a} -> {ys : List a} -> (x :: xs) = (y :: ys) -> xs = ys
tailsSame Refl = Refl

appendLeftCancel : {xs,ys,ys' : List a} -> xs ++ ys = xs ++ ys' -> ys = ys'
appendLeftCancel {xs = []} prf = prf
appendLeftCancel {xs = (x :: xs)} prf = appendLeftCancel {xs} (tailsSame prf)

lengthDrop : (xs,ys : List a) -> drop (length xs) (xs ++ ys) = ys
lengthDrop [] ys = Refl
lengthDrop (x :: xs) ys = lengthDrop xs ys

lengthTake : (xs,ys : List a) -> take (length xs) (xs ++ ys) = xs
lengthTake [] ys = Refl
lengthTake (x :: xs) ys = cong $ lengthTake xs ys

appendRightCancel_lem : (xs,xs',ys : List a) -> xs ++ ys = xs' ++ ys -> length xs = length xs'
appendRightCancel_lem xs xs' ys eq =
  let foo = lengthAppend xs ys
      bar = replace {P = \b => length b = length xs + length ys} eq foo
      baz = trans (sym bar) $ lengthAppend xs' ys
  in plusRightCancel (length xs) (length xs') (length ys) baz

appendRightCancel : {xs,xs',ys : List a} -> xs ++ ys = xs' ++ ys -> xs = xs'
appendRightCancel {xs} {xs'} {ys} eq with (appendRightCancel_lem xs xs' ys eq)
  | lenEq = rewrite sym $ lengthTake xs ys
            in let foo : (take (length xs') (xs ++ ys) = xs') = rewrite eq in lengthTake xs' ys
            in rewrite lenEq in foo

listPartsEqLeft : {xs, xs', ys, ys' : List a} ->
                  length xs = length xs' ->
                  xs ++ ys = xs' ++ ys' ->
                  xs = xs'
listPartsEqLeft {xs} {xs'} {ys} {ys'} leneq appeq =
  rewrite sym $ lengthTake xs ys
  in rewrite leneq
  in rewrite appeq
  in lengthTake xs' ys'

listPartsEqRight : {xs, xs', ys, ys' : List a} ->
                   length xs = length xs' ->
                   xs ++ ys = xs' ++ ys' ->
                   ys = ys'
listPartsEqRight leneq appeq with (listPartsEqLeft leneq appeq)
  listPartsEqRight leneq appeq | Refl = appendLeftCancel appeq


thereInjective : There loc1 = There loc2 -> loc1 = loc2
thereInjective Refl = Refl

injTail : InjList (x :: xs) -> InjList xs
injTail {x} {xs} xxsInj v vloc1 vloc2 = thereInjective $
    xxsInj v (There vloc1) (There vloc2)

splitInorder_lem2 : ((loc1 : Elem v (v :: xs ++ v :: ysr)) ->
                     (loc2 : Elem v (v :: xs ++ v :: ysr)) -> loc1 = loc2) ->
                    Void
splitInorder_lem2 {v} {xs} {ysr} f =
  let
    loc2 = elemAppend {x=v} xs (v :: ysr) Here
  in (\Refl impossible) $ f Here (There loc2)

-- preorderLength and inorderLength could be proven using the bijections
-- between trees and their traversals, but it's much easier to just prove
-- them directly.

preorderLength : (t : Tree a) -> length (preorder t) = size t
preorderLength Tip = Refl
preorderLength (Node l v r) =
  rewrite sym (plusSuccRightSucc (size l) (size r))
  in cong {f=S} $
     rewrite sym $ preorderLength l
     in rewrite sym $ preorderLength r
     in lengthAppend _ _

inorderLength : (t : Tree a) -> length (inorder t) = size t
inorderLength Tip = Refl
inorderLength (Node l v r) =
  rewrite lengthAppend (inorder l) (v :: inorder r)
  in rewrite inorderLength l
  in rewrite inorderLength r in Refl

preInLength : (t : Tree a) -> length (preorder t) = length (inorder t)
preInLength t = trans (preorderLength t) (sym $ inorderLength t)


splitInorder_lem1 : (v : a) ->
                    (xsl, xsr, ysl, ysr : List a) ->
                    (xsInj : InjList (xsl ++ v :: xsr)) ->
                    (ysInj : InjList (ysl ++ v :: ysr)) ->
                    xsl ++ v :: xsr = ysl ++ v :: ysr ->
                    v `Elem` (xsl ++ v :: xsr) ->
                    v `Elem` (ysl ++ v :: ysr) ->
                    xsl = ysl
splitInorder_lem1 v [] xsr [] ysr xsInj ysInj eq locxs locys = Refl
splitInorder_lem1 v [] xsr (v :: ysl) ysr xsInj ysInj eq Here Here with (ysInj v Here (elemAppend (v :: ysl) (v :: ysr) Here))
  splitInorder_lem1 v [] xsr (v :: ysl) ysr xsInj ysInj eq Here Here | Refl impossible
splitInorder_lem1 v [] xsr (y :: ysl) ysr xsInj ysInj eq Here (There loc) with (headsSame eq)
  splitInorder_lem1 v [] xsr (v :: ysl) ysr xsInj ysInj eq Here (There loc) | Refl = absurd $ splitInorder_lem2 (ysInj v)
splitInorder_lem1 v [] xsr (x :: xs) ysr xsInj ysInj eq (There loc) locys with (headsSame eq)
  splitInorder_lem1 v [] xsr (v :: xs) ysr xsInj ysInj eq (There loc) locys | Refl = absurd $ splitInorder_lem2 (ysInj v)
splitInorder_lem1 v (v :: xs) xsr ysl ysr xsInj ysInj eq Here locys = absurd $ splitInorder_lem2 (xsInj v)
splitInorder_lem1 v (x :: xs) xsr [] ysr xsInj ysInj eq (There y) locys with (headsSame eq)
  splitInorder_lem1 v (v :: xs) xsr [] ysr xsInj ysInj eq (There y) locys | Refl = absurd $ splitInorder_lem2 (xsInj v)
splitInorder_lem1 v (x :: xs) xsr (z :: ys) ysr xsInj ysInj eq (There y) locys with (headsSame eq)
  splitInorder_lem1 v (v :: xs) xsr (_ :: ys) ysr xsInj ysInj eq (There y) Here | Refl = absurd $ splitInorder_lem2 (ysInj v)
  splitInorder_lem1 v (x :: xs) xsr (x :: ys) ysr xsInj ysInj eq (There y) (There z) | Refl = cong {f = ((::) x)} $
                           splitInorder_lem1 v xs xsr ys ysr (injTail xsInj) (injTail ysInj) (tailsSame eq) y z

splitInorder_lem3 : (v : a) ->
                    (xsl, xsr, ysl, ysr : List a) ->
                    (xsInj : InjList (xsl ++ v :: xsr)) ->
                    (ysInj : InjList (ysl ++ v :: ysr)) ->
                    xsl ++ v :: xsr = ysl ++ v :: ysr ->
                    v `Elem` (xsl ++ v :: xsr) ->
                    v `Elem` (ysl ++ v :: ysr) ->
                    xsr = ysr
splitInorder_lem3 v xsl xsr ysl ysr xsInj ysInj prf locxs locys with (splitInorder_lem1 v xsl xsr ysl ysr xsInj ysInj prf locxs locys)
  splitInorder_lem3 v xsl xsr xsl ysr xsInj ysInj prf locxs locys | Refl =
     tailsSame $ appendLeftCancel prf

更可怕的外表

injLeft : {l : Tree a} -> {v : a} -> {r : Tree a} ->
          InjTree (Node l v r) -> InjTree l
injLeft {l} {v} {r} injlvr x loc1 loc2 with (injlvr x (OnLeft loc1) (OnLeft loc2))
  injLeft {l = l} {v = v} {r = r} injlvr x loc1 loc1 | Refl = Refl

injRight : {l : Tree a} -> {v : a} -> {r : Tree a} ->
           InjTree (Node l v r) -> InjTree r
injRight {l} {v} {r} injlvr x loc1 loc2 with (injlvr x (OnRight loc1) (OnRight loc2))
  injRight {l} {v} {r} injlvr x loc1 loc1 | Refl = Refl

简单的事实：如果树是单射的，那么它的左右子树也是如此。

t

主要目标！

如果u和t是二叉树，t是单射的，u和t具有相同的预订顺序和顺序遍历，那么{{ 1}}和u相等。

travsDet : (t, u : Tree a) -> InjTree t -> preorder t = preorder u -> inorder t = inorder u -> t = u
-- The base case--both trees are empty
travsDet Tip Tip x prf prf1 = Refl
-- Impossible cases: only one tree is empty
travsDet Tip (Node l v r) x Refl prf1 impossible
travsDet (Node l v r) Tip x Refl prf1  impossible
-- The interesting case. `headsSame presame` proves
-- that the roots of the trees are equal.
travsDet (Node l v r) (Node t y u) lvrInj presame insame with (headsSame presame)
  travsDet (Node l v r) (Node t v u) lvrInj presame insame | Refl =
    let
      foo = elemAppend (inorder l) (v :: inorder r) Here
      bar = elemAppend (inorder t) (v :: inorder u) Here
      inlvrInj = treeInInj _ lvrInj
      intvuInj : (InjList (inorder (Node t v u))) = rewrite sym insame in inlvrInj
      inorderRightSame = splitInorder_lem3 v (inorder l) (inorder r) (inorder t) (inorder u) inlvrInj intvuInj insame foo bar
      preInL : (length (preorder l) = length (inorder l)) = preInLength l
      inorderLeftSame = splitInorder_lem1 v (inorder l) (inorder r) (inorder t) (inorder u) inlvrInj intvuInj insame foo bar
      inPreT : (length (inorder t) = length (preorder t)) = sym $ preInLength t
      preLenlt : (length (preorder l) = length (preorder t))
               = trans preInL (trans (cong inorderLeftSame) inPreT)
      presame' = tailsSame presame
      baz : (preorder l = preorder t) = listPartsEqLeft preLenlt presame'
      quux : (preorder r = preorder u) = listPartsEqRight preLenlt presame'
-- Putting together the lemmas, we see that the
-- left and right subtrees are equal
      recleft = travsDet l t (injLeft lvrInj) baz inorderLeftSame
      recright = travsDet r u (injRight lvrInj) quux inorderRightSame
    in rewrite recleft in rewrite recright in Refl

另一种“没有重复”的概念

如果树中的两个位置不相等，则可能希望说树“没有重复”，因此它们不具有相同的元素。这可以使用NoDups类型表示。

NoDups : Tree a -> Type
NoDups {a} t = (x, y : a) ->
               (loc1 : x `InTree` t) ->
               (loc2 : y `InTree` t) ->
               Not (loc1 = loc2) ->
               Not (x = y)

这足以证明我们需要的原因是有一个程序可以确定树中的两条路径是否相等：

instance DecEq (x `InTree` t) where
  decEq AtRoot AtRoot = Yes Refl
  decEq AtRoot (OnLeft x) = No (\Refl impossible)
  decEq AtRoot (OnRight x) = No (\Refl impossible)
  decEq (OnLeft x) AtRoot = No (\Refl impossible)
  decEq (OnLeft x) (OnLeft y) with (decEq x y)
    decEq (OnLeft x) (OnLeft x) | (Yes Refl) = Yes Refl
    decEq (OnLeft x) (OnLeft y) | (No contra) = No (contra . onLeftInjective)
  decEq (OnLeft x) (OnRight y) = No (\Refl impossible)
  decEq (OnRight x) AtRoot = No (\Refl impossible)
  decEq (OnRight x) (OnLeft y) = No (\Refl impossible)
  decEq (OnRight x) (OnRight y) with (decEq x y)
    decEq (OnRight x) (OnRight x) | (Yes Refl) = Yes Refl
    decEq (OnRight x) (OnRight y) | (No contra) = No (contra . onRightInjective)

这证明Nodups t暗示InjTree t：

noDupsInj : (t : Tree a) -> NoDups t -> InjTree t
noDupsInj t nd x loc1 loc2 with (decEq loc1 loc2)
  noDupsInj t nd x loc1 loc2 | (Yes prf) = prf
  noDupsInj t nd x loc1 loc2 | (No contra) = absurd $ nd x x loc1 loc2 contra Refl

最后，紧接着NoDups t完成工作。

travsDet2 : (t, u : Tree a) -> NoDups t -> preorder t = preorder u -> inorder t = inorder u -> t = u
travsDet2 t u ndt = travsDet t u (noDupsInj t ndt)

Answer 2

想象一下，您有以下预订遍历：a,b,c,d,e,f,g。这告诉你什么？

您知道a是树的根，这来自预订遍历的定义。到目前为止，非常好。

您还知道列表的其余部分是遍历左子树，然后遍历右子树。不幸的是，你不知道分裂的位置。可能是所有这些都属于左侧树，可能是所有这些都属于正确的树，或者b,c向左走，d,e,f,g向右走，依此类推。

如何解决歧义？那么，让我们来看看有序遍历，它的定义属性是什么？左侧子树a中的任何元素将在有序遍历中位于a之前，右侧子树中的任何元素将位于a之后。同样，这是从有序遍历的定义得出的。

所以我们需要做的是看一下有序遍历（让我们说它是c,b,a,d,e,f,g）。我们可以看到b和c出现在a之前，因此它们位于左子树中，而d，e，{{1} }}和f位于正确的子树中。换句话说，有序遍历中的g位置唯一地确定哪些节点将在其左/右子树中。

这很好，因为我们现在可以继续递归地解决这两个子树：预订a /按顺序b,c和预订c,b /按顺序d,e,f,g。

您可以递归地继续此操作，直到所有子树只包含一个元素，其中解决方案非常独特。

因为在每一步我们都可以证明只有一种有效的方法可以继续，结果是一对给定的有序和预先遍历遍历只能属于一棵树。

如果您更喜欢更正式的符号，则可以找到完全相同的证据here。

Answer 3

我问过面试官的一个问题是关于重复的因素。两个＆＃34;不同＆＃34;如果二叉树具有重复元素，则它们可以具有相同的预订顺序和顺序遍历。

例如，请考虑以下情况：

inorder：{12,12} 预购：{12,12}

       12           12 

    /                  \

 12                     12

现在谈到有独特元素的情况。 当我们以递归方式处理问题时，我们总是可以将更大的集合分解为3的元组。让我们说 inorder遍历为{Left，Root，Right}和 pre-order Traversal as {Root，Left，Right}。

当我们从前序遍历中固定Root时，其余的前序遍历应该被认为是两个子部分，其进一步的细节可以从inorder遍历中获得。观察到在每个阶段，我们都试图解决标准的三节点问题：我们可能并不关心有多少＆＃34;子问题＆＃34;每个节点都有，因为我们知道我们稍后会到达那一点。

Answer 4

要创建树，我们需要一个根节点和一个放置顺序。

预订购/后订购提供根节点，而inorder提供放置位置。因此，通过这种推理，我们将需要两个遍历，即预顺序或后顺序以及创建唯一树的顺序。

如果它是BST，则由于我们已经知道节点的放置顺序，因此Preorder或Postorder就足够了。

这是问题的直观推理。