Immutability in C# Part Seven: More on Binary Trees

Lots of good comments on my previous post. To briefly follow up:

• One of the downsides of immutable tree implementations is that usually the tree must be built from the leaves up, which is not always convenient. We'll look at implementations which hide this fact from the user in future posts.

• One smartypants pointed out that sure, this tree can have cycles -- if you have another binary tree implementation that has cycles and then paste such a cyclic tree in. True; what I meant was that using only the stated implementation, it's impossible to create a tree with cycles. If you are hell bent on making a bad tree, you can surely do that. Again, in a future post we'll have a tree which is built up by calling tree "mutating" operations, and those really will be guaranteed acyclic even in a world where there are other bad implementations hanging around.

And finally, one reader correctly identified the problem with my recursive implementation of in-order iteration. A tree of maximum height h ends up allocating O(h) iterator objects. The recursive calls get O(h) deep. If h is very large, this could blow the call stack. And if the tree has n nodes, each with an average height of O(h), then iterating each node will require O(h) recursive calls apiece. Therefore the total time cost in calls for iterating the entire tree is O(n h). 

In a binary tree with n nodes, h is always between log n and n, so that means that this iterator has best case asymptotic performance of O(n log n) and worst case  of O(n²) in time, and uses between O(log n) and O(n) in both call stack and heap space. That's pretty lousy.

A better implementation would be to write a tree traversal algorithm which does not use recursion. Use an explicit stack rather than the call stack:

public static IEnumerable<T> InOrder<T>(this IBinaryTree<T> tree)
{
    IStack<IBinaryTree<T>> stack = Stack<IBinaryTree<T>>.Empty;
    for (IBinaryTree<T> current = tree; !current.IsEmpty || !stack.IsEmpty; current = current.Right)
    {
        while (!current.IsEmpty)
        {
            stack = stack.Push(current);
            current = current.Left;
        }
        current = stack.Peek();
        stack = stack.Pop();
        yield return current.Value;
    }
}

This consumes O(n) time, O(h) heap space, and O(1) call stack space. It is also painful to read and analyze compared to the slow naive implementation.

I would very much like to add new syntax to a hypothetical future version of C# which would be a syntactic sugar for the code above. If, hypothetically, we were prioritizing features for unannounced future versions of C#, unfortunately that one would not be our highest priority, so I wouldn't expect it any time soon. My colleague Wes has written a great article that goes into more details about the problem above and ways we might solve it in the future, so check that out if you want the details.

I have the code for the next few blog posts written, but odds are good that I'm not going to have time to write the surrounding text until after the festive holiday season is over. I hope you have a wonderful time during the remainder of 2007 and we'll see you bright and early in 2008 for more fabulous adventures!

Comments

• Anonymous
  December 19, 2007
  Great series!! Tree traversal can also be done without using stack. For example, http://vadmyst.blogspot.com/2007/09/non-recursive-binary-tree-traversal.html. But, this approach makes tree node more "heavy" as additional information has to be used.

• Anonymous
  December 19, 2007
  In your approach how do you traverse the tree for a second time? All the nodes will be marked as visited. I think what you want to do is store the "visited" information in a separate dictionary object which is owned by the traverser.

• Anonymous
  December 19, 2007
  You could make Visited an integer, incrementing it each time you visit a node. Obviously, it also doesn't work for immutable trees :-)

• Anonymous
  December 19, 2007
  I should say "incrementing it each time you TRAVERSE THE TREE"...

• Anonymous
  December 19, 2007
  Dictionary approach seems to be better, in scenarios when traversing is done only with search purpose. In this case not all nodes will be visited and "visited" counter can be useless.

• Anonymous
  December 20, 2007
  Perhaps a nice feature of a future C# would be to allow a function to delegate the yield ability to some deeper function. That would allow you to write InOrder like this maybe: public static IEnumerable<T> InOrder<T>(this IBinaryTree<T> tree) { yield tree.left.InOrder(); yield return tree.value; yield tree.right.InOrder(); } Of course that's not a trivial feature to implement, but maybe someday C# will have continuations.

• Anonymous
  December 20, 2007
  The InOrder function should be checking against IsEmpty instead of null or Count > 0. And speaking of null, it would be a nice feature to be able to specify that a variable of a class can not be set to null.  This would simplify code so that we would not have to check that a value passed into a method is null because it would not be allowed by the compiler.  This would require being able to identity a default value so that an array could be initialized.  In the case of these immutible data structues, the default would of course by Empty.

• Anonymous
  December 20, 2007
  "I would very much like to add new syntax to a hypothetical future version of C# which would be a syntactic sugar for the code above." It's called F#.

• Anonymous
  December 20, 2007
  Whoops, that's what I get for modifying the code from another project and posting it without compiling it. Thanks for the note.

• Anonymous
  December 24, 2007
  I've a small question unrelated to this immutables series, but I hope is still relevant for this blog: Why did you not include generic properties/indexers in C#? Are you contemplating doing so in the "hypothetical future version of C#".

• Anonymous
  December 27, 2007
  Why do we have to bother about "out-of-stack". Using the stack for recursion in trees are done in functional programming like F#, Lisp all the time. The only datastructure too big to use the stack is lists and similar, which can be several thousand elements. Trees on the other seldom becomes deeper than 100 levels, especially if you use balanced trees like AVL or red-black trees.

• Anonymous
  December 30, 2007
  Stackless in-order binary tree traversal with left, right and parent pointers only Off the top of my head and untested class Node {  Node Parent;  Node Left;  Node Right;  Object Value; } class InOrderTreeWalk {  bool descendLeft;  Node current;  InOrderTreeWalk(Node start) {    this.descendLeft = true;    this.current = start;  }  bool moveNext() {    if(this.current == null) {      return false;    }    if(this.descendLeft) {      this.descendLeft = false;      while(this.current.left != null) {        this.current = this.current.left;      }    }    else {      if(this.current.right != null) {        this.current = this.current.right;        while(this.current.left != null) {          this.current = this.current.left;        }      }      else {        if(this.current.parent != null && this.current == this.current.parent.left) {           this.current = this.current.parent;        }        else {          while(this.current.parent != null && this.current == this.current.parent.right) {            this.current = this.current.parent;          }          if(this.current.parent == null) {            this.current = null;          }        }      }    }    return this.current != null;  }  object getValue() {    return (this.current != null) ? this.current.Value : null;  } }

• Anonymous
  January 01, 2008
  The comment has been removed

• Anonymous
  January 01, 2008
  How are you going to build an immutable binary tree that has parent pointers?

• Anonymous
  January 01, 2008
  Note that "binary tree" and "b tree" are two entirely different data structures. "b tree" is NOT short for "binary tree".

• Anonymous
  January 18, 2008
  For some reason, there's been a lot of buzz lately around immutability in C#. If you're interested in

• Anonymous
  January 19, 2008
  I didn't realize that b-tree was an acronym for another data structure. I meant to say binary tree there. As for building immutable trees with parent pointers, I was not suggesting that, but pointing out that Ifeanyi's binary tree class has parent pointers, whereas Eric's binary tree class does not. The point was that including a parent pointer misleadingly looks like it uses less memory to traverse, when it actually grows at a faster rate than the stack usage.

• Anonymous
  February 13, 2008
  why?        current = stack.Peek();        stack = stack.Pop(); why not?        current = stack.Pop();

• Anonymous
  February 13, 2008
  Because stack.Pop() returns the popped stack.  Remember, it's an immutable stack; popping it does not change the stack, popping it returns you an entirely different object which represents the state of the popped stack. More generally, immutable data structures enable the abstract data type implementer to have every method do exactly one thing, which is good design.  Popping and peeking are two entirely different things. One creates a new data structure, the other inspects an existing data structure. So why should they be the same method? Because mutable stacks are, well, mutable, the implementer of the ADT must somehow work around the inability of callers to reason about the state of the data structure. For this reason, it is very common to see mutable data structures where one method does the work properly done by many different methods, in order to guarantee atomicity. This question has come up in almost every part of this series. See the comments to the other parts for more details.

• Anonymous
  February 13, 2008
  > Why did you not include generic properties/indexers in C#? Is there a compelling customer scenario for doing so which cannot already be easily solved with a generic method? I am not aware of any, but I am happy to hear any such scenario you've got.   > Are you contemplating doing so in the "hypothetical future version of C#". Not at this time, no.  

• Anonymous
  June 17, 2008
  Is there a compelling customer cenario for using properties, that cannot already be solved with a method?? I am not aware of any, but I would be very happy if you would give us one. It's all a matter of syntax. What led to the creation of generic methods would also lead to the creation of generic properties. Yet, I agree that the syntax could become strange...: public interface IServiceContainer    {        bool     Contains<TService>();        TService GetInstanceOf<TService>();        void SetInstanceOf<TService>(TService oInstance);        /* OR /        TService Instance<TService> { get; set;}    }    public interface ISomeService    {    }    public class Usage    {        public Usage()        {            IServiceContainer oContainer = null;// new ServiceContainer();            /             * Configure the container             * ...             /            ISomeService oService = oContainer.GetInstanceOf<ISomeService>();            oContainer.SetInstanceOf<ISomeService>(oService);            / OR */            ISomeService oService = oContainer.InstanceOf<ISomeService>;            oContainer.InstanceOf<ISomeService> = oService;        }    }