2 binary trees are equal or not

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Determine if two binary trees are equal

Got an interview yesterday, a question got me, here it is:

Description

There are 2 binary trees, check if they are equal.

They are equal if and only if tree1->child == tree2->child, and one tree's left and right children can be swapped with each other.

For example:

    5     6
   / \   / \           they are equal.
   1 2   2  1

    5         6
   / \       / \           they are equal.
  1   2     2   1
 /     \   /    / 
3       4 4     3

Any ideas are appreciated.

Answer 1

Equality operators are transitive: If A=B, and B=C, then A=B=C so A=C.

Equality operators are reflexive: A=A, B=B, and C=C no matter what their values are.

Equality operators are symmetric. If A=B, then B=A. (It doesn't matter what order they are in.)

Now, taking a look at the definition they gave you:

A tree is equal to another tree if the children are equal. Let's see. We can assume that the nodes are being compared at the bottom, or else the definition is pretty useless. But they don't bother to tell you how to resolve that comparison, and the whole definition they gave you hinges on it.

In short, it's a crappy question.

Let's see what happens if we decide we want to try and unravel the question, though.

But wait, they also tell you that the two children of any tree can be swapped. This adds the constraint that any tree that is equal to anything else (including itself) must be equal to its mirror image. And any variations of its subtrees' children being swapped.

And remember that this is supposed to be a search tree. Therefore, we can probably assume that two different search trees that are processed by the same algorithm must give the same result if they are equal. So, if we switch around the elements of a tree, then the search time would be affected. So, trees that do not have every node in place are not equal to each other.

Putting that together with the "swappable" property of this equality, we can see it's not a valid definition of equality. (If we try to apply it, then it turns out that only trees that have the same node for every node at a particular level are equal, and only to themselves, which breaks the reflexivity part of an equality operator.)

Answer 2

I don't think this is an unreasonable question. A simple recursive solution is

boolean equals(x, y)
{
  if (x == null)
  {
    return y == null;
  }
  if (y == null)
  {
    return false;
  }
  if (x.val != y.val)
  {
    return false;
  }
  if (equals(x.left, y.left) && equals(x.right, y.right))
  {
    return true;
  }
  if (equals(x.left, y.right) && equals(x.right, y.left))
  {
    return true;
  }
  return false;
}

This can be very expensive, for example in the case when we have two large trees of similar shape where all the non-leaf nodes have the same associated value and the leaf nodes of one are a permutation of the leaf nodes of another.

To get past this you could first of all change left and right as required so that left < right, for some recursive definition of <. This could also be expensive, but much less so than checking every permutation, and I think a choice of definition of < would help. This would then allow you to check for equality with an ordinary definition.

This notion of http://en.wikipedia.org/wiki/Canonicalization followed by ordinary equality also resolves questions about whether you really do have an equivalence relation. An equivalence relation is equivalent to a partition. Ordinary equality is obviously a partition. If you compare x and y by comparing f(x) and f(y) followed by an equivalence relation you have a partition of x and y, and therefore an equivalence relation.

Thinking more about this, I think the way to make either canonicalisation or equality-testing reasonably efficient is to work from the bottom up, annotating each node with a token whose value reflects the result of comparisons with other nodes, so that you can compare nodes, and the subtrees below them, just be comparing tokens.

So the first step for equality is to e.g. use a hash table to annotate each leaf with tokens which are equal only when the values at the leaves are equal. Then, for nodes whose only children are leaves, use e.g. a hash table to assign further tokens so that the tokens in those nodes are equal only when the leaves, if any, beneath those nodes match. Then you can go one more step up, and this time you can compare tokens at child nodes instead of recursing down the tree there. The cost of assigning tokens in this way should be linear in the size of the trees involved. At the top you can compare trees just by comparing the tokens at the root.

Answer 3

If you implement their definition of "equality" with flip-invariance, you will violate the definition of equality. The definition doesn't even make sense, because that's not how binary search trees are equal (unless each node has a pointer to which subtree is "greater" and which is "lesser").

You have two choices of reasonable definitions:

1. topological (flip-agnostic) equivalence (in which case you can't call it a "binary search tree" because it's not sorted):

   tree1==tree2 means set(tree1.children)==set(tree2.children)

2. normal search tree (flip-caring) equivalence:

   tree1==tree2 means list(tree1.children)==list(tree2.children)

For binary trees, the above definitions will work as-written in any language which supports the list and set datatypes (python sets will choke however on unhashable datatypes). Nevertheless, below are some more verbose and ugly C/Java-like definitions:

1. topological equivalence:

   t1==t2 means (t1.left==t2.left and t1.right==t2.right) or (t1.left==t2.right and t1.right==t2.left)

2. sorted tree equivalence:

   t1==t2 means (t1.left==t2.left and t1.right==t2.right)

The definitions above are recursive; that is, they assume equality has been defined for the subtrees and base-cases already, which it has.

---

sidenote:

quote: tree1->child == tree2->child

This is not a valid statement, because a tree node does not have a single child.

Answer 4

Compare trees using canonization approach suggested by @mcdowella. The difference is that my approach doesn't require O(N) additional memory w.r.t number of nodes in the tree:

# in Python
from collections import namedtuple
from itertools import chain

# Tree is either None or a tuple of its value and left, right trees
Tree = namedtuple('Tree', 'value left right')

def canonorder(a, b):
    """Sort nodes a, b by their values.

    `None` goes to the left
    """
    if (a and b and a.value > b.value) or b is None:
        a, b = b, a # swap
    return a, b

def canonwalk(tree, canonorder=canonorder):
    """Yield all tree nodes in a canonical order.

    Bottom-up, smaller children first, None is the smallest
    """
    if tree is not None:
        children = tree[1:]
        if all(t is None for t in children): return # cut None leaves
        children = canonorder(*children)            
        for child in chain(*map(canonwalk, children)):
            yield child
    yield tree 

canonwalk() requires O(N*M) steps and O(log(N)*M) memory to yield all nodes in a tree, where N is total number of nodes, M number of children each node has (it is 2 for binary trees).

canonorder() could be easily generalized for any node representation and any number of children. canonwalk() requires only that a tree can access its immediate children as a sequence.

The comparison function that calls canonwalk():

from itertools import imap, izip_longest

unset = object() 
def cmptree(*trees):
    unequal = False # allow root nodes to be unequal
    # traverse in parallel all trees under comparison
    for nodes in izip_longest(*imap(canonwalk, trees), fillvalue=unset):
        if unequal:
            return False # children nodes are not equal
        if any(t is unset for t in nodes):
            return False # different number of nodes
        if all(t is not None for t in nodes):
            unequal = any(nodes[-1].value != t.value for t in nodes)
        else: # some are None
            unequal = any(t is not None for t in nodes)
    return True # equal

Example

    5         6
   / \       / \           they are equal.
  1   2     2   1
 /     \   /    / 
3       4 4     3

tree1 = Tree(5, 
             Tree(1, 
                  Tree(3, None,None), None), 
             Tree(2, 
                  None, Tree(4, None, None)))
tree2 = Tree(6, 
             Tree(2, Tree(4, None, None), None),
             Tree(1, Tree(3, None, None), None))
print cmptree(tree1, tree2)

Output

True

Answer 5

I read the questions as: given two binary trees, for each depth in the tree, find out whether their children's set are covered in each other.

This can be coded relatively easy.

Answer 6

Solution without recursion in Ruby

def same? top_t1, top_t2
  for_chek << [top_t1, top_t2]   # (1) put task for check into queue

  while t1,t2 = for_check.shift  # (2)
    return false unless t1.children.count == t2.children.count  # generally for non-binary tree, but also needed for controlling of nil children
    break if t1.children.empty?

    t1_children = t1.children.sort # this is sorted arrays
    t2_children = t2.children.sort # of childrens      
    return false unless t1_children == t2_children  # (3)

    0.upto(t1_children.count - 1) do |i|
      for_check << [t1_children[i], t2_children[i]]  # put equivalent child pairs into queue
    end
  end
  return true
end

Ruby syntax tips:

• (1) putting element into array: arr << elem; in this case for_check is array of arrays
• (2) parallel assignment: t1,t2 = [item1, item2]. Same as arr = [item1, item2]; t1 = arr[0]; t2 = arr[1]
• (3) t1_children == t2_children assumed corresponding behavior of == for this kind of objects. More verbose will be t1_children.map { |el| el.val } == t2_children.map { |el| el.val } - here map produces array of vals.